9.1: Ellipses
The National Statuary Hall in Washington, D.C. is an oval-shaped room called a whispering chamber because the shape makes it possible for sound to reflect from the walls in a special way. Two people standing in specific places are able to hear each other whispering even though they are far apart. To determine where they should stand, we will need to better understand ellipses.
Photo by Gary Palmer, Flickr, CC-BY, https://www.flickr.com/photos/gregpalmer/2157517950
An ellipse is a type of conic section, a shape resulting from intersecting a plane with a cone and looking at the curve where they intersect. They were discovered by the Greek mathematician Menaechmus over two millennia ago.
The figure below (Pbroks13 (commons.wikimedia.org/wiki/F...with_plane.svg), “Conic sections with plane”, cropped to show only ellipse and circle by L Michaels, CC BY 3.0) shows two types of conic sections. When a plane is perpendicular to the axis of the cone, the shape of the intersection is a circle. A slightly titled plane creates an oval-shaped conic section called an ellipse.
An ellipse can be drawn by placing two thumbtacks in a piece of cardboard then cutting a piece of string longer than the distance between the thumbtacks. Tack each end of the string to the cardboard, and trace a curve with a pencil held taught against the string. An ellipse is the set of all points where the sum of the distances from two fixed points is constant. The length of the string is the constant, and the two thumbtacks are the fixed points, called foci.
Definition: ELLIPSE Definition AND VOCABULARY
An ellipse is the set of all points \(\;Q\left( {x,y} \right)\) for which the sum of the distance to two fixed points \(F_1 \left( x_1,y_1 \right)\) and \(F_2 \left( x_2,y_2 \right)\), called the foci (plural of focus), is a constant k: \[d\left( {Q,{F_1}} \right) + d\left( {Q,{F_2}} \right) = k\]
- The major axis is the line passing through the foci.
- Vertices are the points on the ellipse which intersect the major axis.
- The major axis length is the length of the line segment between the vertices.
- The center is the midpoint between the vertices (or the midpoint between the foci).
- The minor axis is the line perpendicular to the minor axis passing through the center.
- Minor axis endpoints are the points on the ellipse which intersect the minor axis.
- The minor axis endpoints are also sometimes called co-vertices .
- The minor axis length is the length of the line segment between minor axis endpoints.
Note that which axis is major and which is minor will depend on the orientation of the ellipse. In the ellipse shown at right, the foci lie on the \(y\) axis, so that is the major axis, and the \(x\) axis is the minor axis. Because of this, the vertices are the endpoints of the ellipse on the \(y\) axis, and the minor axis endpoints (co-vertices) are the endpoints on the \(x\) axis.
Ellipses Centered at the Origin
From the definition above we can find an equation for an ellipse. We will find it for a ellipse centered at the origin \(C\left( {0,0} \right)\) with foci at \({F_1}\left( {c,0} \right)\) and \({F_2}\left( { - c,0} \right)\) where \(c > 0\).
Suppose\(\;Q\left( {x,y} \right)\) is some point on the ellipse. The distance from \(F_1\) to \(Q\) is
\[d\left( Q,{F_1} \right) = \sqrt {{{\left( {x - c} \right)}^2} + {{\left( {y - 0} \right)}^2}\;} = \sqrt {{{\left( {x - c} \right)}^2} + {y^2}\;} \nonumber\]
Likewise, the distance from \(F_2\) to \(Q\) is
\[d\left( {Q,{F_2}} \right) = \sqrt {{\left( {x - \left( { - c} \right)} \right)}^2 + {\left( {y - 0} \right)}^2} = \sqrt {{{\left( {x + c} \right)}^2} + {y^2}\;} \nonumber\]
From the definition of the ellipse, the sum of these distances should be constant:
\[d\left( {Q,{F_1}} \right) + d\left( {Q,{F_2}} \right) = k\nonumber\]so that
\[\sqrt {\left( {x - c} \right)^2} + y^2; + \sqrt {{{\left( {x + c} \right)}^2} + {y^2}\;} = k\nonumber\]
If we label one of the vertices \(\left( a,0 \right)\), it should satisfy the equation above since it is a point on the ellipse. This allows us to write \(k\) in terms of \(a\).
\[\sqrt {\left( a - c \right)^2 +0^2} + \sqrt {\left( a + c \right)^2 + 0^2} = k\nonumber\]
\[\left| {a - c} \right| + \left| {a + c} \right| = k\nonumber\] Since \(a > c\), these will be positive
\[(a - c) + (a + c) = k\nonumber\]
\[2a = k\nonumber\]
Substituting that into our equation, we will now try to rewrite the equation in a friendlier form.
\[\sqrt {\left( x - c \right)^2 + y^2} + \sqrt {\left( x + c \right)^2 + y^2} = 2a\nonumber\]Move one radical
\[\sqrt {\left( x - c \right)^2 + y^2} = 2a - \sqrt {\left( x + c \right)^2 + y^2} \nonumber\]Square both sides
\[\left( \sqrt {\left( x - c \right)^2 + y^2 } \right)^2 = \left( 2a - \sqrt {\left( x + c \right)^2 + y^2} \right)^2\nonumber\]Expand
\[\left( x - c \right)^2 + y^2= 4a^2 - 4a\sqrt {\left( x + c \right)^2 + y^2} + \left( x + c \right)^2 + y^2\nonumber\]Expand more
\[x^2 - 2xc + c^2 + y^2 = 4a^2 - 4a\sqrt {\left( x + c \right)^2 + y^2} + x^2 + 2xc + c^2 + y^2\nonumber\]
Combining like terms and isolating the radical leaves
\[4a\sqrt {\left( x + c \right)^2 + y^2} = 4a^2 + 4xc\nonumber\]Divide by 4
\[a\sqrt {\left( x + c \right)^2 + y^2} = a^2 + xc\nonumber\]Square both sides again
\[a^2\left( \left( x + c \right)^2 + y^2 \right) = a^4 + 2a^2xc + x^2c^2\nonumber\]Expand
\[a^2\left( x^2 + 2xc + c^2 + y^2 \right) = a^4 + 2a^2xc + x^2c^2\nonumber\]Distribute
\[a^2x^2 + 2a^2xc + a^2c^2 + a^2y^2 = a^4 + 2a^2xc + x^2c^2\nonumber\]Combine like terms
\[a^2x^2 - x^2c^2 + a^2y^2 =a^4 - a^2c^2\nonumber\]Factor common terms
\[\left( a^2 - c^2 \right)x^2 + a^2y^2 = a^2\left( a^2 - c^2 \right)\nonumber\]
Let \(b^2 = a^2 - c^2\). Since \(a > c\), we know \(b > 0\). Substituting \(b^2\)for \(a^2 - c^2\) leaves
\[b^2x^2 + a^2y^2 = a^2b^2\nonumber\] Divide both sides by \(a^2b^2\)
\[\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\nonumber\]
This is the standard equation for an ellipse. We typically swap \(a\) and \(b\) when the major axis of the ellipse is vertical.
Definition: EQUATION OF AN ELLIPSE CENTERED AT THE ORIGIN IN STANDARD FORM
The standard form of an equation of an ellipse centered at the origin \(C\left( 0,0 \right)\) depends on whether the major axis is horizontal or vertical. The table below gives the standard equation, vertices, minor axis endpoints, foci, and graph for each.
| Major Axis | Horizontal | Vertical |
| Vertices | \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) | \(\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\) |
| Standard Equation | (-\(a\), 0) and (\(a\), 0) | (0, -\(a\)) and (0, \(a\)) |
| Minor Axis Endpoints | (0, -\(b\)) and (0, \(b\)) | (-\(b\), 0) and (\(b\), 0) |
| Foci |
(-\(c\), 0) and (\(c\), 0) where \(b^2 = a^2 - c^2\) |
(0, -\(c\)) and (0, \(c\)) where \(b^2 = a^2 - c^2\) |
| Graph |
Example \(\PageIndex{1}\)
Put the equation of the ellipse \(9{x^2} + {y^2} = 9\) in standard form. Find the vertices, minor axis endpoints, length of the major axis, and length of the minor axis. Sketch the graph, then check using a graphing utility.
Solution
The standard equation has a 1 on the right side, so this equation can be put in standard form by dividing by 9:
\[\dfrac{x^2}{1} + \dfrac{y^2}{9} = 1\nonumber\]
Since the \(y\)-denominator is greater than the \(x\)-denominator, the ellipse has a vertical major axis. Comparing to the general standard form equation \(\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\), we see the value of \(a = \sqrt 9 = 3\) and the value of \(b = \sqrt 1 = 1\).
- The vertices lie on the \(y\)-axis at (0, \(\pm a\)) = (0, \(\pm 3\)).
- The minor axis endpoints lie on the \(x\)-axis at (\(\pm b\), 0) = (\(\pm 1\), 0).
- The length of the major axis is \(2\left( a \right) = 2\left( 3 \right) = 6\).
- The length of the minor axis is \(2\left( b \right) = 2\left( 1 \right) = 2\).
To sketch the graph we plot the vertices and the minor axis endpoints. Then we sketch the ellipse, rounding at the vertices and the minor axis endpoints.
To check on a graphing utility, we must solve the equation for \(y\). Isolating \(y^2\) gives us
\[y^2 = 9\left( 1 - x^2 \right)\nonumber\]
Taking the square root of both sides we get
\[y = \pm 3\sqrt {1 - x^2} \nonumber\]
Under Y= on your graphing utility enter the two halves of the ellipse as \(y = 3\sqrt {1 - x^2} \) and \(y = - 3\sqrt {1 - x^2} \). Set the window to a comparable scale to the sketch with xmin = -5, xmax = 5, ymin= -5, and ymax = 5.
Here’s an example output on a TI-84 calculator:
Sometimes we are given the equation. Sometimes we need to find the equation from a graph or other information.
Example \(\PageIndex{2}\)
Find the standard form of the equation for an ellipse centered at (0,0) with horizontal major axis length 28 and minor axis length 16.
Solution
Since the center is at (0,0) and the major axis is horizontal, the ellipse equation has the standard form \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\). The major axis has length \(2a = 28\) or \(a = 14\). The minor axis has length \(2b = 16\) or b = 8. Substituting gives \[\dfrac{x^2}{16^2} + \dfrac{y^2}{8^2} = 1\text{ or }\dfrac{x^2}{256} + \dfrac{y^2}{64.} = 1\nonumber\]
Exercise \(\PageIndex{1}\)
Find the standard form of the equation for an ellipse with horizontal major axis length 20 and minor axis length 6.
- Answer
-
\(2a = 20\), so \(a =10\). \(2b = 6\), so \(b = 3\). \[\dfrac{x^2}{100} + \dfrac{y^2}{9} = 1\nonumber\]
Example \(\PageIndex{3}\)
Find the standard form of the equation for the ellipse graphed here.
Solution
The center is at (0,0) and the major axis is vertical, so the standard form of the equation will be \(\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\).
From the graph we can see the vertices are (0,4) and (0,-4), giving \(a = 4\).
The minor-axis endpoints are (2,0) and (-2,0), giving \(b = 2\).
The equation will be \[\dfrac{x^2}{2^2} + \dfrac{y^2}{4^2} = 1\text{ or }\dfrac{x^2}{4} + \dfrac{y^2}{16} = 1\nonumber\]
Ellipses Not Centered at the Origin
Not all ellipses are centered at the origin. The graph of such an ellipse is a shift of the graph centered at the origin, so the standard equation for one centered at (\(h\), \(k\)) is slightly different. We can shift the graph right \(h\) units and up \(k\) units by replacing \(x\) with \(x – h\) and \(y\) with \(y – k\), similar to what we did when we learned transformations.
Definition: EQUATION OF AN ELLIPSE CENTERED AT \((h,k)\) IN STANDARD FORM
The standard form of an equation of an ellipse centered at the point C\(\left( {h,k} \right)\) depends on whether the major axis is horizontal or vertical. The table below gives the standard equation, vertices, minor axis endpoints, foci, and graph for each.
| Major Axis | Horizontal | Vertical |
| Standard Equation | \(\dfrac{(x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2} = 1\) | \(\dfrac{(x - h)^2}{b^2} + \dfrac{(y - k)^2}{a^2} = 1\) |
| Vertices | \((h \pm a, k)\) | \((h, k \pm a)\) |
| Minor Axis Endpoints | \((h, k \pm b)\) | \((h \pm b, k)\) |
| Foci |
\((h \pm c, k)\)
where \(b^2 = a^2 - c^2\) |
\((h, k \pm c)\)
where \(b^2 = a^2 - c^2\) |
| Graph |
Example \(\PageIndex{4}\)
Put the equation of the ellipse \(x^2 + 2x + 4y^2 - 24y = - 33\) in standard form. Find the vertices, minor axis endpoints, length of the major axis, and length of the minor axis. Sketch the graph.
Solution
To rewrite this in standard form, we will need to complete the square, twice.
Looking at the \(x\) terms, \(x^2 + 2x\), we like to have something of the form \((x + n)^2\). Notice that if we were to expand this, we’d get \(x^2 + 2nx + n^2\), so in order for the coefficient on \(x\) to match, we’ll need \((x + 1)^2 = x^2 + 2x + 1\). However, we don’t have \(a +1\) on the left side of the equation to allow this factoring. To accommodate this, we will add 1 to both sides of the equation, which then allows us to factor the left side as a perfect square:
\[x^2 + 2x + 1 + 4y^2 - 24y = - 33 + 1\nonumber\]
\[(x + 1)^2 + 4y^2 - 24y = - 32\nonumber\]
Repeating the same approach with the \(y\) terms, first we’ll factor out the 4.
\[4y^2 - 24y = 4(y^2 - 6y)\nonumber\]
Now we want to be able to write \(4\left( y^2 - 6y \right)\) as
\[4(y + n)^2 = 4\left( y^2 + 2ny + n^2 \right)\nonumber\]
For the coefficient of \(y\) to match, \(n\) will have to -3, giving \(4(y - 3)^2 = 4\left( y^2 - 6y + 9 \right) = 4y^2 - 24y + 36\).
To allow this factoring, we can add 36 to both sides of the equation.
\[(x + 1)^2 + 4y^2 - 24y + 36 = - 32 + 36\nonumber\]
\[(x + 1)^2 + 4\left( y^2 - 6y + 9 \right) = 4\nonumber\]
\[(x + 1)^2 + 4\left( y - 3 \right)^2 = 4\nonumber\]
Dividing by 4 gives the standard form of the equation for the ellipse
\[\dfrac{\left( x + 1 \right)^2}{4} + \dfrac{\left( y - 3 \right)^2}{1} = 1\nonumber\]
Since the \(x\)-denominator is greater than the \(y\)-denominator, the ellipse has a horizontal major axis. From the general standard equation \(\dfrac{\left( x - h \right)^2}{a^2} + \dfrac{\left( h - k \right)^2}{b^2} = 1\) we see the value of \(a = \sqrt 4 = 2\) and the value of \(b = \sqrt 1 = 1\).
The center is at (\(h\), \(k\)) = (-1, 3).
The vertices are at (\(h \pm a\), \(k\)) or (-3, 3) and (1,3).
The minor axis endpoints are at (\(h\), \(k \pm b\)) or (-1, 2) and (-1,4).
The length of the major axis is \(2\left( a \right) = 2\left( 2 \right) = 4\).
The length of the minor axis is \(2\left( b \right) = 2\left( 1 \right) = 2\).
To sketch the graph we plot the vertices and the minor axis endpoints. Then we sketch the ellipse, rounding at the vertices and the minor axis endpoints.
Example \(\PageIndex{5}\)
Find the standard form of the equation for an ellipse centered at (-2,1), a vertex at (-2,4) and passing through the point (0,1).
Solution
The center at (-2,1) and vertex at (-2,4) means the major axis is vertical since the x-values are the same. The ellipse equation has the standard form
\[\dfrac{\left( x - h \right)^2}{b^2} + \dfrac{\left( y - k \right)^2}{a^2} = 1\nonumber\]
The value of \(a = 4 - 1 = 3\). Substituting \(a = 3\), \(h = -2\), and \(k = 1\) gives
\[\dfrac{\left( x + 2 \right)^2}{b^2} + \dfrac{\left( y - 1 \right)^2}{3^2} = 1\nonumber\]Substituting for \(x\) and \(y\) using the point (0,1) gives
\[\dfrac{\left( 0 + 2 \right)^2}{b^2} + \dfrac{\left( 1 - 1 \right)^2}{3^2} = 1\nonumber\]
Solving for b gives \(b = 2\).
The equation of the ellipse in standard form is \[\dfrac{\left( x + 2 \right)^2}{2^2} + \dfrac{\left( y - 1 \right)^2}{3^2} = 1\text{ or }\dfrac{\left( x + 2 \right)^2}{4} + \dfrac{\left( y - 1 \right)^2}{9} = 1\nonumber\]
Exercise \(\PageIndex{2}\)
Find the center, vertices, minor axis endpoints, length of the major axis, and length of the minor axis for the ellipse \(\left( x - 4 \right)^2 + \dfrac{\left( y + 2 \right)^2}{4} = 1\).
- Answer
-
Center (4, -2). Vertical ellipse with \(a = 2\), \(b = 1\).
Vertices at (4, -2 \(\pm\) 2) = (4, 0) and (4, -4),
minor axis endpoints at (4 \(\pm\) 1, -2) = (3,-2) and (5,-2),
major axis length 4, minor axis length 2
Bridges with Semielliptical Arches
Arches have been used to build bridges for centuries, like in the Skerton Bridge in England which uses five semielliptical arches for support. Semielliptical arches can have engineering benefits such as allowing for longer spans between supports.
(Maxine Armstrong (commons.wikimedia.org/wiki/F...r,_England.JPG), “Skerton Bridge, Lancaster, England”, CC BY-SA)
Example \(\PageIndex{6}\)
A bridge over a river is supported by a single semielliptical arch. The river is 50 feet wide. At the center, the arch rises 20 feet above the river. The roadway is 4 feet above the center of the arch. What is the vertical distance between the roadway and the arch 15 feet from the center?
Solution
Put the center of the ellipse at (0,0) and make the span of the river the major axis.
Since the major axis is horizontal, the equation has the form \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\).
The value of \(a = \dfrac{1}{2}(50) = 25\) and the value of \(b = 20\), giving \(\dfrac{x^2}{25^2} + \dfrac{y^2}{15^2} = 1\).
Substituting \(x = 15\) gives \(\dfrac{15^2}{25^2} + \dfrac{y^2}{20^2} = 1\). Solving for \(y\), \[y = 20\sqrt {1 - \dfrac{225}{625}} = 16\nonumber\]
The roadway is 20 + 4 = 24 feet above the river. The vertical distance between the roadway and the arch 15 feet from the center is 24 - 16 = 8 feet.
Ellipse Foci
The location of the foci can play a key role in ellipse application problems. Standing on a focus in a whispering gallery allows you to hear someone whispering at the other focus. To find the foci, we need to find the length from the center to the foci, \(c\), using the equation \({b^2} = {a^2} - {c^2}\). It looks similar to, but is not the same as, the Pythagorean Theorem.
Example \(\PageIndex{7}\)
The National Statuary Hall whispering chamber is an elliptical room 46 feet wide and 96 feet long. To hear each other whispering, two people need to stand at the foci of the ellipse. Where should they stand?
Solution
We could represent the hall with a horizontal ellipse centered at the origin. The major axis length would be 96 feet, so \(a = \dfrac{1}{2}(96) = 48\), and the minor axis length would be 46 feet, so \(b = \dfrac{1}{2}(46) = 23\). To find the foci, we can use the equation \(b^2 = a^2 - c^2\).
\[23^2 =48^2 - c^2\nonumber\]
\[c^2 = 48^2 - 23^2\nonumber\]
\[c = \sqrt {1775} \approx \pm 42\text{ ft}\nonumber\]
To hear each other whisper, two people would need to stand 2(42) = 84 feet apart along the major axis, each about 48 – 42 = 6 feet from the wall.
Example \(\PageIndex{8}\)
Find the foci of the ellipse \(\dfrac{\left( x - 2 \right)^2}{4} + \dfrac{\left( y + 3 \right)^2}{29} = 1\).
Solution
The ellipse is vertical with an equation of the form \(\dfrac{\left( x - h \right)^2}{b^2} + \dfrac{\left( y - k \right)^2}{a^2} = 1\).
The center is at (\(h\), \(k\)) = (2, -3). The foci are at (\(h\), \(k \pm c\)).
To find length \(c\) we use \(b^2 = a^2 - c^2\).
Substituting gives \(4 = 29 - c^2\) or \(c = \sqrt {25} = 5\).
The ellipse has foci (2, -3 ± 5), or (2, -8) and (2, 2).
Example \(\PageIndex{9}\)
Find the standard form of the equation for an ellipse with foci (-1,4) and (3,4) and major axis length 10.
Solution
Since the foci differ in the \(x\) -coordinates, the ellipse is horizontal with an equation of the form \(\dfrac{\left( x - h \right)^2}{a^2} + \dfrac{\left( h - k \right)^2}{b^2} = 1\).
The center is at the midpoint of the foci \(\left( \dfrac{x_1 + x_2}{2},\dfrac{y_1 + y_2}{2} \right) = \left( \dfrac{\left( - 1 \right) + 3}{2},\dfrac{4 + 4}{2} \right) = \left( 1,4 \right)\).
The value of \(a\) is half the major axis length: \[a = \dfrac{1}{2}(10) = 5\nonumber\]
The value of \(c\) is half the distance between the foci: \[c = \dfrac{1}{2}(3 - ( - 1)) = \dfrac{1}{2}(4) = 2\nonumber\]
To find length \(b\) we use \({b^2} = {a^2} - {c^2}\). Substituting a and c gives \(b^2 = 5^2 - 2^2\) = 21.
The equation of the ellipse in standard form is \[\dfrac{\left( x - 1 \right)^2}{5^2} + \dfrac{\left( y - 4 \right)^2}{21} = 1\text{ or }\dfrac{\left( x - 1 \right)^2}{25} + \dfrac{\left( y - 4 \right)^2}{21} = 1\nonumber\]
Exercise \(\PageIndex{3}\)
Find the standard form of the equation for an ellipse with focus (2,4), vertex (2,6), and center (2,1).
- Answer
-
Vertex, center, and focus have the same \(x\)-value, so it’s a vertical ellipse.
Using the vertex and center, \[a = 6 – 1 = 5\nonumber\]
Using the center and focus, \[c = 4 – 1 = 3\nonumber\]
\[b^2 = 5^2 - 3^2\quad b = 4\nonumber\]
\[\dfrac{\left( x - 2 \right)^2}{16} + \dfrac{\left( y - 1 \right)^2}{25} = 1\nonumber\]
Planetary Orbits
It was long thought that planetary orbits around the sun were circular. Around 1600, Johannes Kepler discovered they were really elliptical (Technically, they’re approximately elliptical. The orbits of the planets are not exactly elliptical because of interactions with each other and other celestial bodies.). His first law of planetary motion says that planets travel around the sun in an elliptical orbit with the sun as one of the foci. The length of the major axis can be found by measuring the planet’s aphelion, its greatest distance from the sun, and perihelion, its shortest distance from the sun, and summing them together.
Example \(\PageIndex{10}\)
Mercury’s aphelion is 35.98 million miles and its perihelion is 28.58 million miles. Write an equation for Mercury’s orbit.
Solution
Let the center of the ellipse be (0,0) and its major axis be horizontal so the equation will have form \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\).
The length of the major axis is \(2a = 35.98 + 28.58 = 64.56\) giving \(a = 32.28\) and \(a^2 = 1041.9984\).
Since the perihelion is the distance from the focus to one vertex, we can find the distance between the foci by subtracting twice the perihelion from the major axis length: \(2c = 64.56 - 2\left( 28.58 \right) = 7.4\) giving \(c = 3.7\).
Substitution of \(a\) and \(c\) into \(b^2 = a^2 - c^2\) yields \(b^2 = 32.28^2 - 3.7^2 = 1028.3084\).
The equation is \[\dfrac{x^2}{1041.9984} + \dfrac{y^2}{1028.3084} = 1\nonumber\]
Important Topics of This Section
- Ellipse Definition
- Ellipse Equations in Standard Form
- Ellipse Foci
- Applications of Ellipses


11.7: Gauss' Law In Differential Form
Gauss' law is a bit spooky. It relates the field on the Gaussian surface to the charges inside the surface. What if the charges have been moving around, and the field at the surface right now is the one that was created by the charges in their previous locations? Gauss' law --- unlike Coulomb's law --- still works in cases like these, but it's far from obvious how the flux and the charges can still stay in agreement if the charges have been moving around.
For this reason, it would be more physically attractive to restate Gauss' law in a different form, so that it related the behavior of the field at one point to the charges that were actually present at that point. This is essentially what we were doing in the fable of the flea named Gauss: the fleas' plan for surveying their planet was essentially one of dividing up the surface of their planet (which they believed was flat) into a patchwork, and then constructing small a Gaussian pillbox around each small patch. The equation \(E_{\perp}=2\pi k\sigma\) then related a particular property of the local electric field to the local charge density.
In general, charge distributions need not be confined to a flat surface --- life is three-dimensional --- but the general approach of defining very small Gaussian surfaces is still a good one. Our strategy is to divide up space into tiny cubes, like the one on page 621. Each such cube constitutes a Gaussian surface, which may contain some charge. Again we approximate the field using its six values at the center of each of the six sides. Let the cube extend from \(x\) to \(x+dx\), from \(y\) to \(y+dy\), and from \(y\) to \(y+dy\).
The sides at \(x\) and \(x+dx\) have area vectors \(-dydz\hat{\mathbf{x}}\) and \(dydz\hat{\mathbf{x}}\), respectively. The flux through the side at \(x\) is \(-E_x(x)dydz\), and the flux through the opposite side, at \(x+dx\) is \(E_x(x+dx)dydz\). The sum of these is \((E_x(x+dx)-E_x(x))dydz\), and if the field was uniform, the flux through these two opposite sides would be zero. It will only be zero if the field's \(x\) component changes as a function of \(x\). The difference \(E_x(x+dx)-E_x(x)\) can be rewritten as \(dE_x=(dE_x)/(dx)dx\), so the contribution to the flux from these two sides of the cube ends up being
Doing the same for the other sides, we end up with a total flux
where \(dv\) is the volume of the cube. In evaluating each of these three derivatives, we are going to treat the other two variables as constants, to emphasize this we use the partial derivative notation \(\partial\) introduced in chapter 3 ,
and we introduce the notation \(\rho\) (Greek letter rho) for the charge per unit volume, giving
This equation has all the same physical implications as Gauss' law. After all, we proved Gauss' law by breaking down space into little cubes like this. We therefore refer to it as the differential form of Gauss' law, as opposed to \(\Phi=4\pi kq_{in}\), which is called the integral form.
Figure b shows an intuitive way of visualizing the meaning of the divergence.  If the divergence is positive, then the whole cluster will expand, and it will contract its volume if it is placed at a point where the field has \(\rm div \mathbf{E}\lt0\). What if the field is constant? We know based on the definition of the divergence that we should have \(\rm div \mathbf{E}=0\) in this case, and the meter does give the right result: all the balls will feel a force in the same direction, but they will neither expand nor contract.
| Example 36: Divergence of a sine wave |
|---|
|
\(\triangleright\) Figure c shows an electric field that varies as a sine wave. This is in fact what you'd see in a light wave: light is a wave pattern made of electric and magnetic fields. (The magnetic field would look similar, but would be in a plane perpendicular to the page.) What is the divergence of such a field, and what is the physical significance of the result? \(\triangleright\) Intuitively, we can see that no matter where we put the div-meter in this field, it will neither expand nor contract. For instance, if we put it at the center of the figure, it will start spinning, but that's it. Mathematically, let the \(x\) axis be to the right and let \(y\) be up. The field is of the form
\[\begin{equation*} \mathbf{E} = (\text{sin} Kx)\: \hat{\mathbf{y}} , \end{equation*}\]
where the constant \(K\) is not to be confused with Coulomb's constant. Since the field has only a \(y\) component, the only term in the divergence we need to evaluate is
\[\begin{equation*} \mathbf{E} = \frac{\partial E_{y}}{\partial y} , \end{equation*}\]
but this vanishes, because \(E_y\) depends only on \(x\), not \(y\) : we treat \(y\) as a constant when evaluating the partial derivative \(\partial E_{y}/\partial y\), and the derivative of an expression containing only constants must be zero. Physically this is a very important result: it tells us that a light wave can exist without any charges along the way to “keep it going.” In other words, light can travel through a vacuum, a region with no particles in it. If this wasn't true, we'd be dead, because the sun's light wouldn't be able to get to us through millions of kilometers of empty space! |
| Example 37: Electric field of a point charge |
|---|
|
The case of a point charge is tricky, because the field behaves badly right on top of the charge, blowing up and becoming discontinuous. At this point, we cannot use the component form of the divergence, since none of the derivatives are well defined. However, a little visualization using the original definition of the divergence will quickly convince us that div \(E\) is infinite here, and that makes sense, because the density of charge has to be infinite at a point where there is a zero-size point of charge (finite charge in zero volume). At all other points, we have
\[\begin{equation*} \mathbf{E} = \frac{ kq}{ r^2}\hat{\mathbf{r}} , \end{equation*}\]
where \(\hat{\mathbf{r}}=\mathbf{r}/ r=( x\hat{\mathbf{x}}+ y\hat{\mathbf{y}}+ z\hat{\mathbf{z}})/ r\) is the unit vector pointing radially away from the charge. The field can therefore be written as
\[\begin{align*} \mathbf{E} &= \frac{ kq}{ r^3}\hat{\mathbf{r}} \\ &= \frac{ kq( x\hat{\mathbf{x}}+ y\hat{\mathbf{y}}+ z\hat{\mathbf{z}})}{\left( x^2+ y^2+ z^2\right)^\text{3/2}} . \\ \text{The three terms in the divergence are all similar, e.g.,}\\ \frac{\partial E_{x}}{\partial x} &= kq\frac{\partial}{\partial x}\left[\frac{ x}{\left( x^2+ y^2+ z^2\right)^\text{3/2}}\right] \\ &= kq\left[\frac{1}{\left( x^2+ y^2+ z^2\right)^\text{3/2}}-\frac{3}{2}\:\frac{2 x^2}{\left( x^2+ y^2+ z^2\right)^\text{5/2}}\right] \\ &= kq\left( r^{-3}-3 x^2 r^{-5}\right) . \end{align*}\]
Straightforward algebra shows that adding in the other two terms results in zero, which makes sense, because there is no charge except at the origin. |
Gauss' law in differential form lends itself most easily to finding the charge density when we are give the field. What if we want to find the field given the charge density? As demonstrated in the following example, one technique that often works is to guess the general form of the field based on experience or physical intuition, and then try to use Gauss' law to find what specific version of that general form will be a solution.
| Example 38: The field inside a uniform sphere of charge |
|---|
|
\(\triangleright\) Find the field inside a uniform sphere of charge whose charge density is \(\rho\). (This is very much like finding the gravitational field at some depth below the surface of the earth.) \(\triangleright\) By symmetry we know that the field must be purely radial (in and out). We guess that the solution might be of the form
\[\begin{equation*} \mathbf{E} = br^ p\hat{\mathbf{r}} , \end{equation*}\]
where \(r\) is the distance from the center, and \(b\) and \(p\) are constants. A negative value of \(p\) would indicate a field that was strongest at the center, while a positive \(p\) would give zero field at the center and stronger fields farther out. Physically, we know by symmetry that the field is zero at the center, so we expect \(p\) to be positive. As in the example 37 , we rewrite \(\hat{\mathbf{r}}\) as \(\mathbf{r}/ r\), and to simplify the writing we define \(n= p-1\), so
\[\begin{equation*} \mathbf{E} = br^ n\mathbf{r} . \end{equation*}\]
Gauss' law in differential form is
\[\begin{equation*} \rm div \mathbf{E} = 4\pi k\rho , \end{equation*}\]
so we want a field whose divergence is constant. For a field of the form we guessed, the divergence has terms in it like
\[\begin{align*} \frac{\partial E_{x}}{\partial x} &= \frac{\partial}{\partial x}\left( br^{n} x\right) \\ &= b\left( nr^{ n-1}\frac{\partial r}{\partial x} x+r^ n\right) \\ \end{align*}\]
The partial derivative \(\partial r/\partial x\) is easily calculated to be \(x/ r\), so
\[\begin{equation*} \frac{\partial E_{x}}{\partial x} = b\left( nr^{ n-2} x^2+r^ n\right) \end{equation*}\]
Adding in similar expressions for the other two terms in the divergence, and making use of \(x^2+ y^2+ z^2= r^2\), we have
\[\begin{equation*} \rm div \mathbf{E} = b( n+3) r^ n . \end{equation*}\]
This can indeed be constant, but only if \(n\) is 0 or \(-3\), i.e., \(p\) is 1 or \(-2\). The second solution gives a divergence which is constant and zero : this is the solution for the outside of the sphere! The first solution, which has the field directly proportional to \(r\), must be the one that applies to the inside of the sphere, which is what we care about right now. Equating the coefficient in front to the one in Gauss' law, the field is
\[\begin{equation*} \mathbf{E} = \frac{4\pi k\rho}{3} r\:\hat{\mathbf{r}} . \end{equation*}\]
The field is zero at the center, and gets stronger and stronger as we approach the surface. |
Contributors and Attributions
Benjamin Crowell (Fullerton College). Conceptual Physics is copyrighted with a CC-BY-SA license.


6.8: Group Decision-Making
- Understand the advantages and disadvantages of group decision-making, and how managers can improve the quality of group decision-making.
Involving more people in the decision-making process can greatly improve the quality of a manager's decisions and outcomes. However, involving more people can also increase conflict and generate other challenges. We turn now to the advantages and disadvantages of group decision-making.
Advantages of Group Decisions
An advantage to involving groups in decision-making is that you can incorporate different perspectives and ideas. For this advantage to be realized, however, you need a diverse group. In a diverse group, the different group members will each tend to have different preferences, opinions, biases, and stereotypes. Because a variety of viewpoints must be negotiated and worked through, group decision-making creates additional work for a manager, but (provided the group members reflect different perspectives) it also tends to reduce the effects of bias on the outcome. For example, a hiring committee made up of all men might end up hiring a larger proportion of male applicants (simply because they tend to prefer people who are more similar to themselves). But with a hiring committee made up of an equal number of men and women, the bias should be cancelled out, resulting in more applicants being hired based on their qualifications rather than their physical attributes.
Having more people involved in decision-making is also beneficial because each individual brings unique information or knowledge to the group, as well as different perspectives on the problem. Additionally, having the participation of multiple people will often lead to more options being generated and to greater intellectual stimulation as group members discuss the available options. Brainstorming is a process of generating as many solutions or options as possible and is a popular technique associated with group decision-making.
All of these factors can lead to superior outcomes when groups are involved in decision-making. Furthermore, involving people who will be affected by a decision in the decision-making process will allow those individuals to have a greater understanding of the issues or problems and a greater commitment to the solutions.
Disadvantages of Group Decisions
Group decision-making is not without challenges. Some groups get bogged down by conflict, while others go to the opposite extreme and push for agreement at the expense of quality discussions. Groupthink occurs when group members choose not to voice their concerns or objections because they would rather keep the peace and not annoy or antagonize others. Sometimes groupthink occurs because the group has a positive team spirit and camaraderie, and individual group members don’t want that to change by introducing conflict. It can also occur because past successes have made the team complacent.
Often, one individual in the group has more power or exerts more influence than others and discourages those with differing opinions from speaking up ( suppression of dissent ) to ensure that only their own ideas are implemented. If members of the group are not really contributing their ideas and perspectives, however, then the group is not getting the benefits of group decision-making.
How to Form a Quality Group
Effective managers will try to ensure quality group decision-making by forming groups with diverse members so that a variety of perspectives will contribute to the process. They will also encourage everyone to speak up and voice their opinions and thoughts prior to the group reaching a decision. Sometimes groups will also assign a member to play the devil’s advocate in order to reduce groupthink. The devil’s advocate intentionally takes on the role of critic. Their job is to point out flawed logic, to challenge the group’s evaluations of various alternatives, and to identify weaknesses in proposed solutions. This pushes the other group members to think more deeply about the advantages and disadvantages of proposed solutions before reaching and implementing a decision.
The methods we’ve just described can all help ensure that groups reach good decisions, but what can a manager do when there is too much conflict within a group? In this situation, managers need to help group members reduce conflict by finding some common ground—areas in which they can agree, such as common interests, values, beliefs, experiences, or goals. Keeping a group focused on a common goal can be a very worthwhile tactic to keep group members working with rather than against one another. Table 6.3 summarizes the techniques to improve group decision-making.
Table 6.3
| Type of Decision | Technique | Benefit |
|---|---|---|
| Group decisions | Have diverse members in the group. | Improves quality: generates more options, reduces bias |
| Assign a devil’s advocate. | Improves quality: reduces groupthink | |
| Encourage everyone to speak up and contribute. | Improves quality: generates more options, prevents suppression of dissent | |
| Help group members find common ground. | Improves quality: reduces personality conflict |
Table 6.3 (Attribution: Copyright Rice University, OpenStax, under CC-BY 4.0 license)
Conclusion
Decision-making is a crucial daily activity for managers. Decisions range from small and simple, with straightforward answers, to big and complex, with little clarity about what the best choice will be. Being an effective manager requires learning how to successfully navigate all kinds of decisions. Expertise, which develops gradually through learning and experience, generally improves managerial decision-making, but managers rarely rely solely on their own expertise. They also conduct research and collect information from others; they pay attention to ethical implication and their own biases; and they think critically about the information that they have received to make decisions that will benefit the organization and its stakeholders.
- Explain why group decision-making can be more effective than individual decision-making.
- What are some things that can prevent groups from making good decisions?
- As a manager, what can you do to enhance the quality of group decision-making?


10.15: Types of Compensation
- Discuss types of compensation
- Discuss factors affecting compensation
Compensation is generally broken down into two primary types: direct and indirect. Direct compensation consists of monetary payments based on time worked or results achieved, including both base and incentive pay. Indirect compensation includes costs incurred for benefits, including vacations, health insurance, retirement plans, federally required protections (e.g., disability and unemployment insurance), and other employee benefits.
A third type of compensation is intangible. Intangible compensation includes non-financial rewards such as mentoring, awards or badging, recognition, and factors that impact quality of work or work/life balance such as greater autonomy, a flexible work schedule, or the ability to work remotely.
| Direct Compensation | Indirect Compensation | Intangible Compensation |
|---|---|---|
| Base salary/ hourly wage | Retirement Plan | Quality of work/life balance |
| Cash bonuses | Health Insurance | Company values/inspiration |
| Annual incentives | Cars | Future growth opportunity |
Compensation is determined by the interplay of internal and external market and regulatory factors, as summarized below.
Worker Productivity
Theoretically, wages should increase when worker productivity increases. This is an economic calculation based on the market structure, costs of production, and market price and quantity.
Supply and Demand for Labor
Theoretically, wages should increase when the demand for labor exceeds supply or, as is the case currently, unemployment is at a historical low. There may also be geographic variances in wages based on the local/regional availability of and competition for labor.
Legislation
As discussed in Fair Labor Standards Act, laws such as the FSLA—including state and local variations—set a minimum standard for compensation.
Labor Unions
Wages are one of a union’s mandatory collective bargaining issues. As we’ll discuss in Module 13: Union–Management Relations, employers who are subject to collective bargaining agreements tend to have higher wage and benefit costs than similar employers without union representation.
Cost of Living
An increase in inflation results in a corresponding loss of purchasing power. Theoretically, wages should increase when the cost of living increases so workers are able to maintain their standard of living. There is also a geographic element to this factor. We see this in the variation in minimum wages set in cities and localities where the cost of living is relatively high.
Industry Ability to Pay
There is a significant gap in what a technology company like Facebook can pay and what a non-profit, small business or start-up can afford to pay. The compensation differential may be offset by the sense of purpose or the opportunity to build a small business.
Compensation Policy
An employer’s position relative to the competition or market will affect the wages it offers. Generally, employers will take one of three positions: lead, lag or match. If an employer’s policy is to lead the market, they will pay relatively higher wages than the competition. The rationale is that they will be able to attract a higher caliber of employees and see an offset in increased productivity and reduced training and turnover costs. Similarly, the face value employee cost savings of a “lag” practice may be offset by higher recruiting, training and turnover costs. A match policy is simply to set compensation at the market rate.
- " Managing Reward: Why Line Managers are the Vital Link ." go2HR. Accessed August 22, 2019. ↵
Contributors and Attributions
- Authored by : rawpixel. Provided by : Pixabay. Located at : pixabay.com/photos/american-analyzing-brainstorming-3748708/. License : CC0: No Rights Reserved . License Terms : Pixabay License


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abuse
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physical, sexual, or emotional harm inflicted on others
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Calgary Family Assessment Model (CFAM)
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model used to assess a family’s strengths, resources, problems, and illness suffering by studying the family’s structure, development, and function
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Calgary Family Intervention Model
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model based on CFAM; provides a framework for nurses to use to have therapeutic conversations that target the family’s functioning as a system using three domains of family function: cognitive, behavioral, and affective
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developmental crisis
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a crisis resulting from predictable change due to normal growth and development
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Duvall’s Family Development Theory
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a theory of family development that asserts that families must move through certain stages, in order, and in each stage attain certain developmental tasks
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ecomap
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visual assessment tool that illustrates the relationships between a family and its social network
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emotional abuse
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a behavior that harms a person’s self-worth or emotional well-being
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family
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a group of two or more people (one of whom is the householder) related by birth, marriage, or adoption and residing together (including related subfamily members)
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family group
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any two or more people (not necessarily including a householder) residing together and related by birth, marriage, or adoption
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family health assessment
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a process of collecting data from family members and organizing this data
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family household
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a group of two or more people (one of whom is the householder) related by birth, marriage, or adoption, including any unrelated people (unrelated subfamily members and/or secondary individuals) who may reside with them
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family nursing
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a nursing discipline that involves caring for the family as one unit or group of people
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family unit
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a household maintained by a householder who is in a family; includes any unrelated people (unrelated subfamily members and/or secondary individuals) who may reside together
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general systems theory
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an interdisciplinary conceptual framework focusing on wholeness, patterns, relationship, hierarchical order, integration, and organization of phenomena
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genogram
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a diagram that depicts family ties through birth and marriage
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intimate partner violence (IVP)
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abuse that is caused by someone who is or has been romantically involved with the victim
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neglect
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the failure to meet the basic physiological and emotional needs of children and vulnerable adults
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physical abuse
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the use of physical force to cause physical injury
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sexual abuse
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pressuring or forcing a person to engage in sexual acts
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situational crisis
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a crisis that can either be predictable, such as a divorce, or unpredictable, such as a natural disaster or family death
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transactional model
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a family nursing model that looks at family processes and the family’s relationships or its transactions with other institutions


14.8: Health Policy and the Future of Healthcare
Policymaking is an important part of public health. Typically we think of policies as laws, regulations, or funding requirements created and enforced by governments. In the U.S., public health is governed at the local, state, and national levels. The interplay between different bureaucracies - at times cooperative and at others in opposition with each other - can actually have just as much of an impact on health policies as the politicians (and interest groups) who propose them (Shi & Singh, 2017).
A prime example of health policies are those associated with smoking. At the national level, the sale of tobacco products to anyone under 21 years of age is prohibited, and the products must all carry a warning label from the Surgeon General. Additionally, the Food and Drug Administration (FDA) has authority to regulate tobacco products (Office on Smoking and Health, CDC, 2023). At the state level, California was one of the first states to ban smoking in indoor public spaces and workplaces in 1995. Smoking had been previously banned in public schools and daycare centers, and taxes are levied on cigarette purchases to fund tobacco-prevention programs through the CA Department of Public Health. In the 2000’s, additional state legislation made smoking illegal on all government property - including state universities and community colleges - and it is also illegal to smoke in a car with a minor present (California Air Resources Board, n.d.). Cities can also enact local smoking bans to protect its residents from environmental tobacco smoke (secondhand smoke), as the city of Burbank in California has done. In Burbank, smoking is prohibited in or around any city facility or park, on any public transit or at any station, on sidewalks in the downtown area, outdoor cafes and event venues, as well as service areas (such as a delivery dock for a restaurant, or a ticket line for a concert) (Burbank Municipal Code, 2024).
Health policies also include the appropriation of tax revenue to fund specific government agencies. The Department of Health and Human Services (HHS) budget for 2024 was $2.37 trillion, which includes funding for Medicare/Medicaid, the Centers for Disease Control and Prevention (CDC), National Institute of Health (NIH), Food and Drug Administration (FDA), Health Resources and Services Administration (HRSA), Substance Abuse and Mental Health Administration (SAMHSA), and Indian Health Service (IHS) as well as other agencies (USAspending.gov, 2024).


23.5: Transformers
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The electric power generated in power stations is transmitted using high-voltage transmission lines, typically with voltages above \(300000\text{V}\) for long distances. However, that voltage is not usable in our households, as our appliances expect a voltage around \(120\text{V}\) (or \(\220\text{V}\) in Europe). Transformers use electromagnetic induction to transform one alternating voltage into another. Figure \(\PageIndex{1}\) illustrates a transformer.
The transformer has two coils, the “primary” and the “secondary”, with different numbers of loops, \(N_p\) , and \(N_s\) , respectively. The coils are wrapped around an iron core, which can transmit the magnetic flux generated in the primary coil to the secondary coil. In the transformer, an alternating voltage, \(\Delta V_p\) , is applied to the primary coil, and transformed into the desired voltage, \(\Delta V_s\) , in the secondary coil.
The current in the primary coil creates a magnetic field. Those field lines are transmitted by the iron core into the second coil. A voltage is only induced in the secondary coil if the magnetic flux through the secondary coil changes with time. Thus, transformers only work with alternating voltages, so that the magnetic field created by the primary coil changes continuously. Both coils will have the same magnetic flux, \(\Phi_B\) , through them, since they have the same area. The voltage in the primary coil is given by Faraday’s Law:
\[\begin{aligned} \Delta V_p = N_p \frac{d\Phi_B}{dt}\end{aligned}\]
as is the voltage in the secondary coil:
\[\begin{aligned} \Delta V_s = N_s \frac{d\Phi_B}{dt}\end{aligned}\]
Since the flux (and thus its time-derivative) are the same in both coils, we can isolate the time-derivative in each equation to obtain the relationship between the voltages in the two coils:
\[\begin{aligned} \frac{\Delta V_p}{N_p}&=\frac{\Delta V_s}{N_s}\\[4pt] \therefore \Delta V_s &= \frac{N_p}{N_s}\Delta V_p\end{aligned}\]
Thus, with a transformer, one simply needs to set the ratio of the number of loops in each coil in order to transform one voltage into another.
Exercise \(\PageIndex{1}\)
Which coil in Figure \(\PageIndex{1}\) has the highest voltage?
- The one with the most loops.
- The one with the least loops.
- Answer
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Exercise \(\PageIndex{2}\)
Which coil in Figure \(\PageIndex{1}\) will have the highest current?
- The one with the most loops.
- The one with the least loops.
- Not enough information to tell.
- Answer
-
Example \(\PageIndex{1}\)
A power plant produces energy at rate of \(P=150\text{kW}\) , and wishes to transmit this power as efficiently as possible to a town. The power lines between the power plant and the town have a resistance of \(R=0.5\Omega\) . Compare the amount of power dissipated in the transmission lines depending on whether the power is transmitted through a voltage of \(300000\text{V}\) or \(300\text{V}\) .
Solution
We model the transmission of power from the power plant to the town using the circuit shown in Figure \(\PageIndex{2}\).
We do not know the resistance of the town, but we can still calculate the power that is dissipated in the transmission lines that have a total resistance of \(R=0.5\Omega\) . The power plant produces power, \(P\) , and transmits it through the lines at a potential difference, \(\Delta V\) , resulting in a current, \(I\) :
\[\begin{aligned} P &= I\Delta V\\[4pt] \therefore I &= \frac{P}{\Delta V}\end{aligned}\]
The current, \(I\) , will dissipate power in the lines at a rate of:
\[\begin{aligned} P_{line} = I^2 R = \frac{P^2}{\Delta V^2}R\end{aligned}\]
With the two different voltages, this corresponds to:
\[\begin{aligned} P_{line}&=\frac{P^2}{\Delta V^2}R=\frac{(150\times 10^{3}\text{W})^2}{(300000\text{V})^2}(0.5\Omega)=0.1\text{W}\\[4pt] P_{line}&=\frac{P^2}{\Delta V^2}R=\frac{(150\times 10^{3}\text{W})^2}{(300\text{V})^2}(0.5\Omega)=125000\text{W}\\[4pt]\end{aligned}\]
Thus, when the power is transmitted at low voltage, more than 80% is dissipated in the transmission lines, whereas an insignificant fraction is dissipated when the power is transmitted at high voltage. This is why we need transformers.


5.4: Complex and Vector-Valued Functions on \(E^{1}\)
The theorems of §§2-3 fail for complex and vector-valued functions (see Problem 3 below and Problem 2 in §3). However, some analogues hold. In a sense, they even are stronger, for, unlike the previous theorems, they do not require the existence of a derivative on an entire interval \(I \subseteq E^{1},\) but only on \(I-Q\), where \(Q\) is a countable set, one contained in the range of a sequence, \(Q \subseteq\left\{p_{m}\right\}.\) (We henceforth presuppose §9 of Chapter 1.)
In the following theorem, due to N. Bourbaki, \(g : E^{1} \rightarrow E^{*}\) is extended real while \(f\) may also be complex or vector valued. We call it the finite increments law since it deals with "finite increments" \(f(b)-f(a)\) and \(g(b)-g(a) .\) Roughly, it states that \(\left|f^{\prime}\right| \leq g^{\prime}\) implies a similar inequality for increments.
Let \(f : E^{1} \rightarrow E\) and \(g : E^{1} \rightarrow E^{*}\) be relatively continuous and finite on a closed interval \(I=[a, b] \subseteq E^{1},\) and have derivatives with \(\left|f^{\prime}\right| \leq g^{\prime},\) on \(I-Q\) where \(Q \subseteq\left\{p_{1}, p_{2}, \ldots, p_{m}, \ldots\right\}.\) Then
\[|f(b)-f(a)| \leq g(b)-g(a).\]
The proof is somewhat laborious, but worthwhile. (At a first reading, one may omit it, however.) We outline some preliminary ideas.
Given any \(x \in I,\) suppose first that \(x>p_{m}\) for at least one \(p_{m} \in Q.\) In this case, we put
\[Q(x)=\sum_{p_{m}<x} 2^{-m};\]
here the summation is only over those \(m\) for which \(p_{m}<x.\) If, however, there are no \(p_{m} \in Q\) with \(p_{m}<x,\) we put \(Q(x)=0.\) Thus \(Q(x)\) is defined for all \(x \in I.\) It gives an idea as to "how many" \(p_{m}\) (at which \(f\) may have no derivative) precede \(x.\) Note that \(x<y\) implies \(Q(x) \leq Q(y)\). (Why?) Also,
\[Q(x) \leq \sum_{m=1}^{\infty} 2^{-m}=1.\]
Our plan is as follows. To prove (1), it suffices to show that for some fixed \(K \in E^{1},\) we have
\[(\forall \varepsilon>0) \quad|f(b)-f(a)| \leq g(b)-g(a)+K \varepsilon,\]
for then, letting \(\varepsilon \rightarrow 0,\) we obtain (1). We choose
\[K=b-a+Q(b), \text { with } Q(x) \text { as above. }\]
Temporarily fixing \(\varepsilon>0,\) let us call a point \(r \in I\) "good" iff
\[|f(r)-f(a)| \leq g(r)-g(a)+[r-a+Q(r)] \varepsilon\]
and "bad" otherwise. We shall show that \(b\) is "good." First, we prove a lemma.
Every "good" point \(r \in I(r<b)\) is followed by a whole interval \((r, s), r<s \leq b,\) consisting of "good" points only.
- Proof
-
First let \(r \notin Q,\) so by assumption, \(f\) and \(g\) have derivatives at \(r,\) with
\[\left|f^{\prime}(r)\right| \leq g^{\prime}(r).\]
Suppose \(g^{\prime}(r)<+\infty.\) Then (treating \(g^{\prime}\) as a right derivative) we can find \(s>r\) \((s \leq b)\) such that, for all \(x\) in the interval \((r, s),\),
\[\left|\frac{g(x)-g(r)}{x-r}-g^{\prime}(r)\right|<\frac{\varepsilon}{2} \quad\text {(why?);}\]
similarly for \(f.\) Multiplying by \(x-r,\) we get
\[\begin{aligned}\left|f(x)-f(r)-f^{\prime}(r)(x-r)\right|<(x-r) \frac{\varepsilon}{2} \text { and } \\ \left|g(x)-g(r)-g^{\prime}(r)(x-r)\right|<(x-r) \frac{\varepsilon}{2}, \end{aligned}\]
and hence by the triangle inequality (explain!),
\[|f(x)-f(r)| \leq\left|f^{\prime}(r)\right|(x-r)+(x-r) \frac{\varepsilon}{2}\]
and
\[g^{\prime}(r)(x-r)+(x-r) \frac{\varepsilon}{2}<g(x)-g(r)+(x-r) \varepsilon.\]
Combining this with \(\left|f^{\prime}(r)\right| \leq g^{\prime}(r),\) we obtain
\[|f(x)-f(r)| \leq g(x)-g(r)+(x-r) \varepsilon \text { whenever } r<x<s.\]
Now as \(r\) is "good," it satisfies \((2) ;\) hence, certainly, as \(Q(r) \leq Q(x)\),
\[|f(r)-f(a)| \leq g(r)-g(a)+(r-a) \varepsilon+Q(x) \varepsilon \text { whenever } r<x<s.\]
Adding this to (3) and using the triangle inequality again, we have
\[|f(x)-f(a)| \leq g(x)-g(a)+[x-a+Q(x)] \varepsilon \text { for all } x \in(r, s).\]
By definition, this shows that each \(x \in(r, s)\) is "good," as claimed. Thus the lemma is proved for the case \(r \in I-Q,\) with \(g^{\prime}(r)<+\infty\).
The cases \(g^{\prime}(r)=+\infty\) and \(r \in Q\) are left as Problems 1 and 2. \(\quad \square\)
We now return to Theorem 1.
Proof of Theorem 1. Seeking a contradiction, suppose \(b\) is "bad," and let \(B \neq \emptyset\) be the set of all "bad" points in \([a, b].\) Let
\[r=\inf B, \quad r \in[a, b].\]
Then the interval \([a, r)\) can contain only "good" points, i.e., points \(x\) such that
\[|f(x)-f(a)| \leq g(x)-g(a)+[x-a+Q(x)] \varepsilon.\]
As \(x<r\) implies \(Q(x) \leq Q(r),\) we have
\[|f(x)-f(a)| \leq g(x)-g(a)+[x-a+Q(r)] \varepsilon \text { for all } x \in[a, r).\]
Note that \([a, r) \neq \emptyset,\) for by (2), \(a\) is certainly "good' (why?), and so Lemma 1 yields a whole interval \([a, s)\) of "good" points contained in \([a, r).\)
Letting \(x \rightarrow r\) in (4) and using the continuity of \(f\) at \(r,\) we obtain (2). Thus \(r\) is "good" itself. Then, however, Lemma 1 yields a new interval \((r, q)\) of "good" points. Hence \([a, q)\) has no "bad" points, and so \(q\) is a lower bound of the set \(B\) of "bad" points in \(I\), contrary to \(q>r=\operatorname{glb} B\). This contradiction shows that \(b\) must be "good," i.e.,
\[|f(b)-f(a)| \leq g(b)-g(a)+[b-a+Q(b)] \varepsilon.\]
Now, letting \(\varepsilon \rightarrow 0,\) we obtain formula (1), and all is proved. \(\quad \square\)
If \(f : E^{1} \rightarrow E\) is relatively continuous and finite on \(I=[a, b] \subseteq\) \(E^{1},\) and has a derivative on \(I-Q,\) then there is a real \(M\) such that
\[|f(b)-f(a)| \leq M(b-a) \text { and } M \leq \sup _{t \in I-Q}\left|f^{\prime}(t)\right|.\]
- Proof
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Let
\[M_{0}=\sup _{t \in I-Q}\left|f^{\prime}(t)\right|.\]
If \(M_{0}<+\infty,\) put \(M=M_{0} \geq\left|f^{\prime}\right|\) on \(I-Q,\) and take \(g(x)=M x\) in Theorem 1. Then \(g^{\prime}=M \geq\left|f^{\prime}\right|\) on \(I-Q,\) so formula (1) yields (5) since
\[g(b)-g(a)=M b-M a=M(b-a).\]
If, however, \(M_{0}=+\infty,\) let
\[M=\left|\frac{f(b)-f(a)}{b-a}\right|<M_{0}.\]
Then (5) clearly is true. Thus the required \(M\) exists always. \(\quad \square\)
Let \(f\) be as in Corollary 1. Then \(f\) is constant on \(I\) iff \(f^{\prime}=0\) on \(I-Q.\)
- Proof
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If \(f^{\prime}=0\) on \(I-Q,\) then \(M=0\) in Corollary 1, so Corollary 1 yields, for any subinterval \([a, x](x \in I),|f(x)-f(a)| \leq 0;\) i.e., \(f(x)=f(a)\) for all \(x \in I.\) Thus \(f\) is constant on \(I.\)
Conversely, if so, then \(f^{\prime}=0,\) even on all of \(I. \quad \square\)
Let \(f, g : E^{1} \rightarrow E\) be relatively continuous and finite on \(I=[a, b],\) and differentiable on \(I-Q.\) Then \(f-g\) is constant on \(I\) iff \(f^{\prime}=g^{\prime}\) on \(I-Q.\)
- Proof
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Apply Corollary 2 to the function \(f-g. \quad \square\)
We can now also strengthen parts (ii) and (iii) of Corollary 4 in §2.
Let \(f\) be real and have the properties stated in Corollary 1. Then
(i) \(f \uparrow\) on \(I=[a, b]\) iff \(f^{\prime} \geq 0\) on \(I-Q ;\) and
(ii) \(f \downarrow\) on \(I\) iff \(f^{\prime} \leq 0\) on \(I-Q\).
- Proof
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Let \(f^{\prime} \geq 0\) on \(I-Q.\) Fix any \(x, y \in I(x<y)\) and define \(g(t)=0\) on \(E^{1}.\) Then \(\left|g^{\prime}\right|=0 \leq f^{\prime}\) on \(I-Q .\) Thus \(g\) and \(f\) satisfy Theorem 1 (with their roles reversed on \(I,\) and certainly on the subinterval \([x, y].\) Thus we have
\[f(y)-f(x) \geq|g(y)-g(x)|=0, \text { i.e., } f(y) \geq f(x) \text { whenever } y>x \text { in } I,\]
so \(f \uparrow\) on \(I\).
Conversely, if \(f \uparrow\) on \(I,\) then for every \(p \in I,\) we must have \(f^{\prime}(p) \geq 0,\) for otherwise, by Lemma 1 of §2, \(f\) would decrease at \(p.\) Thus \(f^{\prime} \geq 0\), even on all of \(I,\) and (i) is proved. Assertion (ii) is proved similarly. \(\quad \square\)


7.11C: Bacterial Transduction
Transduction is the process by which DNA is transferred from one bacterium to another by a virus.
LEARNING OBJECTIVES
Differentiate between generalized and specialized transduction
Key Takeaways
Transduction
Transduction is the process by which DNA is transferred from one bacterium to another by a virus. It also refers to the process whereby foreign DNA is introduced into another cell via a viral vector. Transduction does not require physical contact between the cell donating the DNA and the cell receiving the DNA (which occurs in conjugation), and it is DNAase resistant (transformation is susceptible to DNAase). Transduction is a common tool used by molecular biologists to stably introduce a foreign gene into a host cell’s genome.
Transduction : Transduction is the process by which DNA is transferred from one bacterium to another by a virus. It also refers to the process whereby foreign DNA is introduced into another cell via a viral vector.
When bacteriophages (viruses that infect bacteria) infect a bacterial cell, their normal mode of reproduction is to harness the replicational, transcriptional, and translation machinery of the host bacterial cell to make numerous virions, or complete viral particles, including the viral DNA or RNA and the protein coat.
Transduction is especially important because it explains one mechanism by which antibiotic drugs become ineffective due to the transfer of antibiotic-resistance genes between bacteria. In addition, hopes to create medical methods of genetic modification of diseases such as Duchenne/Becker Muscular Dystrophy are based on these methodologies.
The Lytic Cycle and the Lysogenic Cycle
Transduction happens through either the lytic cycle or the lysogenic cycle. If the lysogenic cycle is adopted, the phage chromosome is integrated (by covalent bonds) into the bacterial chromosome, where it can remain dormant for thousands of generations. If the lysogen is induced (by UV light for example), the phage genome is excised from the bacterial chromosome and initiates the lytic cycle, which culminates in lysis of the cell and the release of phage particles. The lytic cycle leads to the production of new phage particles which are released by lysis of the host.
Transduction is a method for transferring genetic material. The packaging of bacteriophage DNA has low fidelity and small pieces of bacterial DNA, together with the bacteriophage genome, may become packaged into the bacteriophage genome. At the same time, some phage genes are left behind in the bacterial chromosome.
There are generally three types of recombination events that can lead to this incorporation of bacterial DNA into the viral DNA, leading to two modes of recombination.
Generalized transduction is the process by which any bacterial gene may be transferred to another bacterium via a bacteriophage, and typically carries only bacterial DNA and no viral DNA. In essence, this is the packaging of bacterial DNA into a viral envelope. This may occur in two main ways, recombination and headful packaging.
If bacteriophages undertake the lytic cycle of infection upon entering a bacterium, the virus will take control of the cell’s machinery for use in replicating its own viral DNA. If by chance bacterial chromosomal DNA is inserted into the viral capsid which is usually used to encapsulate the viral DNA, the mistake will lead to generalized transduction.
If the virus replicates using “headful packaging,” it attempts to fill the nucleocapsid with genetic material. If the viral genome results in spare capacity, viral packaging mechanisms may incorporate bacterial genetic material into the new virion.
The new virus capsule, now loaded with part bacterial DNA, continues to infect another bacterial cell. This bacterial material may become recombined into another bacterium upon infection.
Fates of DNA Inserted into the Recipient Cell
When the new DNA is inserted into this recipient cell it can fall to one of three fates: the DNA will be absorbed by the cell and be recycled for spare parts; if the DNA was originally a plasmid, it will recirculate inside the new cell and become a plasmid again; if the new DNA matches with a homologous region of the recipient cell’s chromosome, it will exchange DNA material similar to the actions in conjugation. This type of recombination is random and the amount recombined depends on the size of the virus being used.
Specialized transduction is the process by which a restricted set of bacterial genes are transferred to another bacterium. The genes that get transferred (donor genes) depend on where the phage genome is located on the chromosome. Specialized transduction occurs when the prophage excises imprecisely from the chromosome so that bacterial genes lying adjacent to the prophage are included in the excised DNA. The excised DNA is then packaged into a new virus particle, which can then deliver the DNA to a new bacterium, where the donor genes can be inserted into the recipient chromosome or remain in the cytoplasm, depending on the nature of the bacteriophage.
When the partially encapsulated phage material infects another cell and becomes a “prophage” (is covalently bonded into the infected cell’s chromosome), the partially coded prophage DNA is called a “heterogenote. ” Example of specialized transduction is λ phages in Escherichia coli , which was discovered by Esther Lederberg.
Key Points
- Transduction does not require physical contact between the cell donating the DNA and the cell receiving the DNA (which occurs in conjugation), and it is DNAase resistant.
- Transduction happens through either the lytic cycle or the lysogenic cycle.
- Transduction is especially important because it explains one mechanism by which antibiotic drugs become ineffective due to the transfer of antibiotic-resistance genes between bacteria.
Key Terms
- lytic cycle : The normal process of viral reproduction involving penetration of the cell membrane, nucleic acid synthesis, and lysis of the host cell.
- lysogenic cycle : A form of viral reproduction involving the fusion of the nucleic acid of a bacteriophage with that of a host, followed by proliferation of the resulting prophage.
- transduction : Transduction is the process by which DNA is transferred from one bacterium to another by a virus.


2.5: Cecil Calvert and Lord Baltemore (1605–1675)
From his father George Calvert, Cecil Calvert inherited the title of Lord Baltimore and the charter from King Charles I to establish a colony at the Province of Maryland, comprising ten to twelve million acres of land in what is now the state of Maryland. Calvert governed the colony from England, sending his Instructions to the Colonists by Lord Baltimore to his brother Leonard, who served as the colony’s first governor. Calvert’s instructions served as the foundation for Maryland’s laws.
Throughout his proprietorship, Cecil Calvert fostered religious tolerance in the colony of Maryland. After Leonard’s death, Calvert commissioned a Protestant, William Stone, to serve as governor. He gave Stone a new law to be voted on by the Maryland Assembly, a law that came to be known as the Act of Toleration. This new law allowed colonists freedom of worship in any Christian faith, provided they maintained loyalty to Cecil Calvert and Maryland’s government.


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7.0: Video Introduction to Chapter 7
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Video introduction to the topics to be covered in this chapter: continuous motion in one and two dimensions (discussing multiple coordinate systems), one-dimensional non-continuous motion, dependent and relative motion.
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7.1: One-Dimensional Continuous Motion
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Quantifying one-dimensional continuous motion of a particle in terms of its position, velocity and acceleration as functions of time. Includes worked examples.
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7.2: One-Dimensional Noncontinuous Motion
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Describing one-dimensional noncontinuous motion of an object in terms of its position, velocity, and acceleration as functions of time. Includes worked examples.
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7.3: Two-Dimensional Kinematics with Rectangular Coordinates
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Analyzing two-dimensional kinematics problems in the Cartesian plane; separating position, velocity, and acceleration into xxx and yyy components. Includes worked examples.
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7.4: Two-Dimensional Kinematics with Normal-Tangential Coordinates
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Overview of the normal-tangential coordinate system. Calculating velocity and acceleration of an object in motion, in terms of the normal and tangential directions. Includes worked examples.
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7.5: Two-Dimensional Motion with Polar Coordinates
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Overview of the polar coordinate system. Calculating the velocity and acceleration of a object in motion in a polar coordinate system as functions of time, in terms of angle to the xxx-axis and distance from the origin. Includes worked examples.
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7.6: Dependent Motion Systems
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Analyzing dependent motion systems and determining their constraints. Includes worked example.
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7.7: Relative Motion Systems
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Analysis of relative motion in multi-object systems, in one and two dimensions. Includes worked examples.
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7.8: Chapter 7 Homework Problems
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5.1: Introduction
Learning Objectives
- Explore the nurse’s role and legal and ethical responsibilities when providing care to clients with mental health disorders
- Describe standards of care for nurses
- Discuss the elements of a malpractice lawsuit
- Explain patient rights based on the Patient Self-Determination Act
The Code of Ethics for Nurses With Interpretive Statements states, “The nurse promotes, advocates for, and protects the rights, health, and safety of the patient.” [1] This chapter describes nurses’ legal and ethical responsibilities while protecting the safety and rights of clients receiving mental health care.
- American Nurses Association. (2015). Code of ethics for nurses with interpretive statements. American Nurses Association. https://www.nursingworld.org/practice-policy/nursing-excellence/ethics/code-of-ethics-for-nurses/coe-view-only / ↵


8.9: Conjunct Verb
Conjunct Verb with करना Video
Lecture: Conjunct Verb
In English, one can use a noun or an adjective like a verb. To use a noun or adjective as a verb in Hindi, one can combine verbs such as करना and होना with them. These combinations cannot be used to create a verb which already exists. However, Hindi does not always have a single verb equivalent to English verbs: such as to close (बंद करना).
| Adjectival conjunct verb | Nominal conjunct verb | |
| Forms | Adjective + करना | Noun + करना |
| Properties |
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Examples of Adjectival Conjunct Verbs
- मैंने अपना कमरा साफ़ किया| “I cleaned my room.”
- तुम अपना घर गंदा करते हो| “You make your house dirty.”
- वे कार्यक्रम की तैयारी कर रहे हैं| “They are preparing for the program.”
Examples of Nominal Conjunct Verbs
Nominal conjunct verbs can be further divided into two categories:
(a) one that requires a connecter such as kaa/kii .
Examples of nouns that require connectors:
- हमने उसकी मदद की| “We helped him/her.”
- राहुल ने गाड़ी की मरम्मत की| “Rahul fixed the car.”
- छात्र हिन्दी का अभ्यास कर रहे हैं| “Students are practicing Hindi.”
- राहुल ने घर की सफ़ाई की| “Rahul cleaned a house.”
(b) one that does not require any connector.
Examples of nouns that do not require connectors:
- राम ने मुझे क्षमा किया| “Rahul forgave me.”
- बच्चे शोर कर रहे हैं| “Children are making noise.”
- मैंने अपनी माँ को फ़ोन किया| “I called my mother.”
Conjunct Verb with होना
Conjunct verbs with होना are intransitive verbs and hence the subject must not appear with the -ne (-ने) marker when the conjunct verb comes in the perfective tense (or expresses a completed action). All the conjunct verbs with करना also have a construction with होना in which the doer of the action is usually not expressed.Adjective + होना
Examples of adjectival conjunct verbs
- मेरा कमरा साफ़ हुआ | “My room got cleaned.”
- उसका कमरा गंदा हुआ | “His house became dirty.”
- कार्यकर्म की तैयारी हो रही है | “Preparation for the program is taking place.”
Noun + होना
Like करना conjunct verb, Nominal conjunct verbs with होना can also be further dived into two categories
(a) one that requires connecter such as kaa/kii.
Examples of nouns that require connectors:
- उसकी मदद हुई | “His/her help happened (literally)./ S/he was helped.”
- गाड़ी की मरम्मत हुई | “The fixing of the car happened (literally)./ The car was fixed.”
- हिन्दी का अभ्यास हो रहा है | “Practice of Hindi is happening (literally)./Urdu is being practiced.”
(b) one that does not require any connector.
Examples of nouns that do not require connectors:
- यहाँ शोर हो रहा है | “It is noisy here.”
- कार्यकर्म खत्म हो गया | “The program ended.”
- भूल हो गयी | “The mistake happened.”
Activity 1
Please write two sentences with each of the following conjunct verbs.
| फ़ैसला करना “To decide” |
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| साफ़ करना “To clean” |
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| वापस करना “To return” |
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| ख़त्म करना “To finish” |
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| कम करना “To reduce” |
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| ठीक करना “To fix up” |
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| तैयार करना “To prepare” |
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| शुरू करना “To start” |
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| मरम्मत करना “To repair” |
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| x की मदद करना “To help X” |
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| x का इंतज़ार करना “To wait for X” |
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| x की कोशिश करना “To try X” |
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| x का अभ्यास करना “To practice X” |
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Activity 2
Please write a response of these following personalized questions and then record yourself.
https://openbooks.lib.msu.edu/ll151/?p=1367
धन्यवाद !


3.9: Interpretation of Coordinate Independence
This section discusses some of the issues that arise in the interpretation of coordinate independence. It can be skipped on a first reading.
Is Coordinate Independence Obvious?
One often hears statements like the following from relativists: “Coordinate independence isn’t really a physical principle. It’s merely an obvious statement about the relationship between mathematics and the physical universe. Obviously the universe doesn’t come equipped with coordinates. We impose those coordinates on it, and the way in which we do so can never be dictated by nature.” The impressionable reader who is tempted to say, “Ah, yes, that is obvious,” should consider that it was far from obvious to Newton (“Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external . . . ”), nor was it obvious to Einstein. Levi-Civita nudged Einstein in the direction of coordinate independence in 1912. Einstein tried hard to make a coordinate-independent theory, but for reasons described in section 3.6, he convinced himself that that was a dead end. In 1914-15 he published theories that were not coordinate independent, which you will hear relativists describe as “obvious” dead ends because they lack any geometrical interpretation. It seems to me that it takes a highly refined intuition to regard as intuitively “obvious” an issue that Einstein struggled with like Jacob wrestling with Elohim.
Is Coordinate Independence Trivial?
It has also been alleged that coordinate independence is trivial. To gauge the justice of this complaint, let’s distinguish between two reasons for caring about coordinate independence:
- Coordinate independence tells us that when we solve problems, we should avoid writing down any equations in notation that isn’t manifestly intrinsic, and avoid interpreting those equations as if the coordinates had intrinsic meaning. Violating this advice doesn’t guarantee that you’ve made a mistake, but it makes it much harder to tell whether or not you have.
- Coordinate independence can be used as a criterion for judging whether a particular theory is likely to be successful.
Nobody questions the first justification. The second is a little trickier. Laying out the general theory systematically in a 1916 paper, 14 Einstein wrote “The general laws of nature are to be expressed by equations which hold good for all the systems of coordinates, that is, are covariant with respect to any substitutions whatever (generally covariant).” In other words, he was explaining why, with hindsight, his 1914-1915 coordinate-dependent theory had to be a dead end.
The only trouble with this is that Einstein’s way of posing the criterion didn’t quite hit the nail on the head mathematically. As Hilbert famously remarked, “Every boy in the streets of Göttingen understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians.” What Einstein had in mind was that a theory like Newtonian mechanics not only lacks coordinate independence, but would also be impossible to put into a coordinate-independent form without making it look hopelessly complicated and ugly, like putting lipstick on a pig. But Kretschmann showed in 1917 that any theory could be put in coordinate independent form, and Cartan demonstrated in 1923 that this could be done for Newtonian mechanics in a way that didn’t come out particularly ugly. Physicists today are more apt to pose the distinction in terms of “background independence” (meaning that a theory should not be phrased in terms of an assumed geometrical background) or lack of a “prior geometry” (meaning that the curvature of spacetime should come from the solution of field equations rather than being imposed by fiat). But these concepts as well have resisted precise mathematical formulation. 15 My feeling is that this general idea of coordinate independence or background independence is like the equivalence principle: a crucial conceptual principle that doesn’t lose its importance just because we can’t put it in a mathematical box with a ribbon and a bow. For example, string theorists take it as a serious criticism of their theory that it is not manifestly background independent, and one of their goals is to show that it has a background independence that just isn’t obvious on the surface.
Coordinate Independence as a Choice of Gauge
It is instructive to consider coordinate independence from the point of view of a field theory. Newtonian gravity can be described in three equivalent ways: as a gravitational field g , as a gravitational potential \(\phi\), or as a set of gravitational field lines. The field lines are never incident on one another, and locally the field satisfies Poisson’s equation. The electromagnetic field has polarization properties different from those of the gravitational field, so we describe it using either the two fields ( E, B ), a pair of potentials, 16 or two sets of field lines. There are similar incidence conditions and local field equations (Maxwell’s equations). Gravitational fields in relativity have polarization properties unknown to Newton, but the situation is qualitatively similar to the two foregoing cases. Now consider the analogy between electromagnetism and relativity. In electromagnetism, it is the fields that are directly observable, so we expect the potentials to have some extrinsic properties. We can, for example, redefine our electrical ground, \(\Phi \rightarrow \Phi + C\), without any observable consequences. As discussed in more detail in Section 5.6, it is even possible to modify the electromagnetic potentials in an entirely arbitrary and nonlinear way that changes from point to point in spacetime. This is called a gauge transformation. In relativity, the gauge transformations are the smooth coordinate transformations. These gauge transformations distort the field lines without making them cut through one another.
Note
There is the familiar electrical potential \(\phi\), measured in volts, but also a vector potential A , which you may or may not have encountered. Briefly, the electric field is given not by \(− \nabla \phi\) but by \(− \nabla \phi − \frac{\partial A}{\partial t}\), while the magnetic field is the curl of A . This is introduced at greater length in section 4.2.
References
15 Giulini, “Some remarks on the notions of general covariance and background independence,” arxiv.org/abs/gr-qc/0603087v1


5.3: Functions of Communication Theory
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While theories in many disciplines can be hard for some to understand, in a field like Communication, our theories are important to understand because they directly impact our daily lives. In this respect, they serve several functions in guiding our communication.
Communication Theory and You
In elementary school you might have believed in cooties. Or, you might have believed that if a boy was mean to a girl, he must have liked her, and vice versa. In Jr. High and High School, finding a date to the homecoming or prom could be one of the most intimidating things to do. Now, in college, the dating world has once again evolved. The ambiguity between what defines a date and a friendly night out can be frustrating for some and exciting for others. Regardless, when situations like these appear, it is easy to seek advice from friends about the situation, ask a parent, or search the web for answers. Each of these resources will likely provide theories about functioning in relationships that you can choose to use or dismiss when clarifying the relationship’s dynamic situation. What are some theories you’ve heard about how to communicate in cross-gendered relationships?
The first function theories serve is that they help us organize and understand our communication experiences. We use theories to organize a broad range of experiences into smaller categories by paying attention to “common features” of communication situations (Infante, Rancer & Womack). How many times have you surfed the internet and found articles or quizzes on relationships and what they mean for different genders? Deborah Tannen, author of You Just Don’t Understand: Women and Men in Conversations , argues that men and women talk in significantly different ways and for significantly different reasons. Of course, these differences cannot be applied to all men and women. But, theories on gender communication help us organize and understand the talk of the different genders in a more simplified context so we can understand general patterns of communication behavior. This helps us make appropriate decisions in gendered communication situations.
A second function of theories is that they help us choose what communicative behaviors to study. Theories guide where we choose to look, what we look at, and how we look at communicative phenomenon. Remember back to Chapter 1 where we defined communication study. Theories focus our attention on certain aspects of that definition. If you find that Tannen’s theories regarding how men and women talk differ from your own perceptions, or that they’re outdated, you might choose to more closely study the talk or non-verbals of men and women to see if you can rectify the difference in theoretical perspectives. You likely already do this on a personal level. Googling something as simple as “how to act in a relationship,” will lead you to hundreds of websites and articles breaking down the dynamics of relationships depending on one’s gender. Likewise, if you want to persuade someone to do something for you, you probably have a theory about what strategies you can use to get them to do what you want. Your theory guides how you approach your persuasive attempts, and what you look for to see if you were successful or not.
A third function of theories is that they help us broaden our understanding of human communication. Scholars who study communication share theories with one another online, through books, journal articles, and at conferences. The sharing of theories generates dialogue, which allows us to further refine the theories developed in this field. Tannen’s book allowed the public to re-think the personal theories they had about the communication of men and women. With the opportunity to find countless theories through new books, magazines, the Internet, and TV shows, the general public has the opportunity to find theories that will influence how they understand and communicate in the world. But, are these theories valid and useful? It’s likely that you discuss your personal theories of communication with others on a regular basis to get their feedback.
A fourth function of theories is that they help us predict and control our communication. When we communicate, we try to predict how our interactions will develop so we can maintain a certain level of control. Imagine being at a party and you want to talk to someone that you find attractive. You will use some sort of theory about how to talk to others to approach this situation in order to make it more successful. As in all situations, the better your theoretical perspectives, the better chances for success when communicating. While theories do not allow us to predict and control communication with 100% certainty, they do help us function in daily interactions at a more predictable and controlled level. Notice that when you are successful, or unsuccessful, in your interactions, you use this information to assess and refine your own theoretical perspectives.
A fifth function of theories is that they help us challenge current social and cultural realities by providing new ways of thinking and living. People sometimes make the mistake of assuming that the ways we communicate are innate rather than learned. This is not true. In order to challenge the communicative norms we learn, people use critical theories to ask questions about the status quo of human communication, particularly focusing on how humans use communication to bring advantage and privilege to particular people or groups. For example, Tannen argues that when men listen to women express their troubles, they listen with the purpose of wanting to provide a fix, or give advice. Tannen argues that many times, women are not looking for advice or a fix, but rather empathy or sympathy from their male conversational partners. With this understanding, it’s possible to begin teaching men new strategies for listening in cross-gendered conversations that serve to build stronger communication ties. Critical theories challenge our traditional theoretical understandings, providing alternative communicative behaviors for social change.
While theories serve many useful functions, these functions don’t really matter if we do not have well-developed theories that provide a good representation of how our world works. While we all form our personal theories through examining our experiences, how are communication theories developed?
Contributions and Affiliations
- Survey of Communication Study. Authored by : Scott T Paynton and Linda K Hahn. Provided by : Humboldt State University. Located at : en.wikibooks.org/wiki/Survey_of_Communication_Study/Preface. License : CC BY-SA: Attribution-ShareAlike


19.7: Spotlight on ... Delivery/Public Speaking
By the end of this section, you will be able to:
- Implement various technologies effectively to address an audience, matching the capacities of each to the rhetorical situation.
- Apply conventions of speech delivery, such as voice control, gestures, and posture.
- Identify and show awareness of cultural considerations.
Think of a speech you have seen or heard, either in person, on television, or online. Was the speech delivered well, or was it poorly executed? What aspects of the performance make you say that? Both good and poor delivery of a speech can affect the audience’s opinion of the speaker and the topic. Poor delivery may be so distracting that even the message of a well-organized script with strong information is lost to the audience.
Speaking Genres: Spoken Word, Pulpit, Youtube, Podcast, Social Media
The world today offers many new (and old) delivery methods for script writing. While the traditional presidential address or commencement speech on a stage in front of a crowd of people is unlikely to disappear, newer script delivery methods are now available, including many that involve technology. From YouTube, which allows anyone to upload videos, to podcasts, which provide a platform for anyone, celebrities and noncelebrities alike, to produce a radio-like program, it seems that people are finding new ways to use technology to enhance communication. Free resources such as YouTube Studio ( https://openstax.org/r/ YouTube_Studio) and the extension TubeBuddy ( https://openstax.org/r/TubeBuddy ) can be a good starting place to learn to create these types of media.
Voice Control
Whether the method is old or new, delivering communication in the speaking genre relies not only on words but also on the way those words are delivered. Remember that voice and tone are important in establishing a bond with your audience, helping them feel connected to your message, creating engagement, and facilitating comprehension. Vocal delivery includes these aspects of speech:
- Rate of speech refers to how fast or slow you speak.  In addition, you can vary your rate, speeding up or slowing down to increase tension, emphasize a point, or create a dramatic effect.
- Volume refers to how loudly or softly you speak. As with rate, you do not want to be too loud or too soft. Too soft, and your speech will be difficult or impossible to hear, even with amplification; too loud, and it will be distracting or even painful for the audience. Ideally, you should project your voice, speaking from the diaphragm, according to the size and location of the audience and the acoustics of the room. You can also use volume for effect; you might use a softer voice to describe a tender moment between mother and child or a louder voice to emphatically discuss an injustice.
- Pitch refers to how high or low a speaker’s voice is to listeners. A person’s vocal pitch is unique to that person, and unlike the control a speaker has over rate and volume, some physical limitations exist on the extent to which individuals can vary pitch. Although men generally have lower-pitched voices than women, speakers can vary their pitch for emphasis. For example, you probably raise your pitch naturally at the end of a question. Changing pitch can also communicate enthusiasm or indicate transition or closure.
- Articulation refers to how clearly a person produces sounds. Clarity of voice is important in speech; it determines how well your audience understands what you are saying. Poor articulation can hamper the effect of your script and even cause your audience to feel disconnected from both you and your message. In general, articulation during a presentation before an audience tends to be more pronounced and dramatic than everyday communication with individuals or small groups. When presenting a script, avoid slurring and mumbling. While these may be acceptable in informal communication, in presented speech they can obscure your message.
- Fluency refers to the flow of speech. Speaking with fluency is similar to reading with fluency. It’s not about how fast you can speak, but how fluid and meaningful your speech is. While inserting pauses for dramatic effect is perfectly acceptable, these are noticeably different from awkward pauses that result from forgetting a point, losing your place, or becoming distracted. Practicing your speech can greatly reduce fluency issues. A word on verbal fillers , those pesky words or sounds used to fill a gap or fluency glitch: utterances such as um , ah , and like detract from the fluency of your speech, distract the audience from your point, and can even reduce your credibility. Again, practice can help reduce their occurrence, and self-awareness can help you speak with more fluency.
Gestures and Expressions
Beyond vocal delivery, consider also physical delivery variables such as gestures and facial expressions . While not all speech affords audiences the ability to see the speaker, in-person, online, and other forms of speech do. Gestures and facial expressions can both add to and detract from effective script delivery, as they can help demonstrate emotion and enthusiasm for the topic. Both have the ability to emphasize points, enhance tone, and engage audiences.
Eye contact is another form of nonverbal, physical communication that builds community, communicates comfort, and establishes credibility. Eye contact also can help hold an audience’s attention during a speech. It is advisable to begin your speech by establishing eye contact with the audience. One idea is to memorize your opening and closing statements to allow you to maintain consistent eye contact during these important sections of the script and strengthen your connection with the audience.
Although natural engagement through gestures, facial expressions, and eye contact can help an audience relate to a presenter and even help establish community and trust, these actions also can distract audiences from the content of the script if not used purposefully. In general, as with most delivery elements, variation and a happy medium between “too much” and “too little” are key to an effective presentation. Some presenters naturally have more expressive faces, but all people can learn to control and use facial expressions and gestures consciously to become more effective speakers. Practicing your speech in front of a mirror will allow you to monitor, plan, and practice these aspects of physical delivery
Figure \(19.11\) Body language such as facial expressions and gestures can help your audience interpret another level of meaning when you address them publicly. (credit: “Mamallapuram, Indian Dance Festival, Bharatanatyam dancer” by Arian Zwegers/flickr, CC BY 2.0)
Posture and Movement
Other physical delivery considerations include posture and movement. Posture is the position of the body. If you have ever been pestered to “stand up straight,” you were being instructed on your posture. The most important consideration for posture during a speech is that you look relaxed and natural. You don’t want to be slumped over and leaning on the podium or lectern, but you also don’t want a stiff, unnatural posture that makes you look stilted or uncomfortable. In many speeches, the speaker’s posture is upright as they stand behind a podium or at a microphone, but this is not always the case. Less formal occasions and audiences may call for movement of the whole body. If this informality fits your speech, you will need to balance movement with the other delivery variables. This kind of balance can be challenging. You won’t want to wander aimlessly around the stage or pace back and forth on the same path. Nor will you want to shuffle your feet, rock, or shift your weight back and forth. Instead, as with every other aspect of delivery, you will want your movements to be purposeful, with the intention of connecting with or influencing your audience. Time your movements to occur at key points or transitions in the script.
Cultural Considerations
Don’t forget to reflect on cultural considerations that relate to your topic and/or audience. Cultural awareness is important in any aspect of writing, but it can have an immediate impact on a speech, as the audience will react to your words, gestures, vocal techniques, and topic in real time. Elements that speakers don’t always think about—including gestures, glances, and changes in tone and inflection—can vary in effectiveness and even politeness in many cultures. Consideration for cultural cues may include the following:
- Paralanguage: voiced cultural considerations, including tone, language, and even accent.
- Kinesics: body movements and gestures that may include facial expressions. Often part of a person’s subconscious, kinesics can be interpreted in various ways by members of different cultures. Body language can include posture, facial expressions (smiling or frowning), and even displays of affection.
- Proxemics: interpersonal space that regulates intimacy. Proxemics might indicate how close to an audience a speaker is located, whether the speaker moves around, and even how the speaker greets the audience.
- Chronemics: use of time. Chronemics refers to the duration of a script.
- Appearance: clothing and physical appearance. The presentation of appearance is a subtle form of communication that can indicate the speaker’s identity and can be specific to cultures.
Stage Directions
You can think proactively about ways to enhance the delivery of your script, including vocal techniques, body awareness, and cultural considerations. Within the draft of your script, create stage directions . An integral part of performances such as plays and films, stage directions can be as simple as writing in a pause for dramatic effect or as complicated as describing where and how to walk, what facial expressions to make, or how to react to audience feedback.
Look at this example from the beginning of the student sample. Stage directions are enclosed in parentheses and bolded.
Several years ago, I sat in the waiting area of a major airport, trying to ignore the constant yapping of a small dog cuddled on the lap of a fellow passenger. An airline rep approached the woman and asked the only two questions allowed by law. (high-pitched voice with a formal tone) “Is that a service animal? (pause) What service does it provide for you?”
(bold, defiant, self-righteous tone) “Yes. It keeps me from having panic attacks,” the woman said defiantly, and the airline employee retreated. (move two steps to the left for emphasis)
Shortly after that, another passenger arrived at the gate. (spoken with authority) She gripped the high, stiff handle on the harness of a Labrador retriever that wore a vest emblazoned with the words “The Seeing Eye.” (speed up speech and dynamic of voice for dramatic effect) Without warning, the smaller dog launched itself from its owner’s lap, snarling and snapping at the guide dog. (move two steps back to indicate transition)
Your Turn
Now it’s your turn. Using the principle illustrated above, create stage directions for your script. Then, practice using them by presenting your script to a peer reviewer, such as a friend, family member, or classmate. Also consider recording yourself practicing your script. Listen to the recording to evaluate it for delivery, fluency, and vocal fillers. Remember that writing is recursive: you can make changes based on what works and what doesn’t after you implement your stage directions. You can even ask your audience for feedback to improve your delivery.
Podcast Publication
If possible, work with your instructor and classmates to put together a single podcast ( https://openstax.org/r/ podcast) or a series of podcasts ( https://openstax.org/r/podcasts ) according to the subject areas of the presentations. The purpose of these podcasts should be to invite and encourage other students to get involved in important causes. Work with relevant student organizations on campus to produce and publicize the podcasts for maximum impact. There are many free resources for creating podcasts, including Apple’s GarageBand ( https://openstax.org/r/GarageBand ) and Audacity ( https://openstax.org/r/Audacity ).