7.5.2: Midlatitudes to the Poles
Beginning along the northwest coast of the United States (coastal Oregon and Washington) and the west coast of Canada (British Columbia) we see a narrow strip of high annual precipitation. The large amount of precipitation is due to the orographic uplift of west-to-east flowing air masses from the ocean. These moist maritime polar air masses are forced to rise up the windward side of north-south oriented mountain chains inducing them to drop copious amounts of precipitation. In many places the amount of precipitation can exceed 100 inches rivaling the amounts found in the tropical rain forests. A short distance inland on the leeward side shows much less precipitation. The lee side of a mountain sits in the "rain shadow".
Traveling down the coast we experience a decrease in precipitation once we get to the southwestern coast of North America around southern California. The annual rainfall map doesn’t gives us a very good picture of what the actual precipitation regime is like in this part of the world though. Examine the climograph for San Francisco, California (Figure \(\PageIndex{1}\)) . San Francisco is located in what’s called the dry summer subtropical climate. The dry summer is due to the presence of the subtropical high that migrates into this region during the summer. The subsidence of the subtropical high keeps air from rising off the surface and dries the air through adiabatic warming. Additionally, as maritime air streams toward land it passes over the cold California Current. The cold ocean current chills the air near the surface and steepens the environmental lapse rate of temperature in the air mass making it stable (mT s ). The combined effect of these factors inhibits precipitation formation and thus creates the dry summer typical of this region. Winter is the wet season as subtropical high pressure slides south and the polar front with its accompanying cyclonic activity migrates into this region.
Proceeding into the desert of the southwestern portion of the United States a noticeable decrease in precipitation occurs. The meager amount of precipitation found in this region is in part due to the lack of available water for evaporation. The desert is located on the leeward side of mountains which also accounts for part of the dryness. Additionally, a high–level high pressure pattern over the western United States is responsible for the summer-time dry conditions. At the surface, heating creates a shallow region of low pressure. Though air can be uplifted by convection and the convergence of air into the low at the surface, an upper–level high helps to suppress cloud development and precipitation.
Moving eastward across the western states one experiences an alternating pattern of lower and higher amounts of precipitation. This pattern is due to the orographic uplift on the wet windward slopes and then drier leeward slopes. The fact that one is a great distance from the source region for moist air also decreases the likelihood for precipitation in the intermountain west.
High annual precipitation and humid conditions prevail in the southeastern part of the United States. The source region for maritime tropical air is located over the Gulf of Mexico and the Caribbean. When this air mass streams over land, convective uplift can create significant precipitation. Along the southeast coast the air mass travels over the warm Gulf Stream which increases the instability of the air. The effect of location with respect to the source region can be seen in the pattern of precipitation. Precipitation decreases as one moves inland toward the northwest. Figure \(\PageIndex{3}\) illustrates the effect of location on the seasonal variation of precipitation as one moves away from the ocean. As one moves poleward the precipitation decreases and becomes more seasonal.
A moderate amount of precipitation occurs in the central US increasing toward the east and northeast coast of the United States and Canada. This region is dominated by the polar front and cyclonic activity. The clash of cold air masses from the north and warmer ones to the south creates frontal uplift and precipitation. Precipitation decreases northward through Canada as air temperature and its associated saturation point decreases.
Moving through the subarctic, Arctic and Polar geographical zones precipitation decreases as temperatures decrease along with the saturation point of air masses. The increasing influence of the polar high helps to suppress precipitation as well. Coastal areas may receive slightly more precipitation due to their proximity to water.
Europe and Asia experience broadly similar patterns as North America, with precipitation decreasing toward the interior of the continent and as one proceeds toward higher latitudes. But there are some very distinct differences too. One of the most notable differences is the penetration of moist conditions farther into the western side of the Eurasian continent than along the western side of North America. While much of western Europe lies in the domain of the westerlies as does the United States, the orientation of the mountains in Europe and North America are quite different. The mountains in North America, especially along the west coast tend to run North-to-South perpendicular to the prevailing winds. This orientation forces air to rise, creating precipitation on the western slope and drier "rain shadow" conditions on the eastern slopes. Mountains tend to be oriented in an east-to-west fashion and thus do not create a barrier to marine time air masses pushed along by the westerlies. This creates moister condition further inland in Europe than North America.
The sheer size of the Eurasian continent creates much drier conditions in the interior than what occurs in North America. Far removed from the ocean, with the imposing barrier of the Himalaya Mountains, interior deserts like the Gobi and Taklimakan, can be found. Like North America, drier conditions become more prevalent as one proceeds towards bone-chilling temperatures near the Arctic Circle and beyond.


5.1: Ratios
A ratio is a comparison of two numerical quantities in the same units that is usually expressed as a fraction.
Examples:
- The ratio of 2 meters to 7 meters is \(\frac{2m}{7m}=\frac{2}{7}\).
-
There are 200,000 gallons of water in a rectangular basin and 400,000 gallons of water in a cylindrical basin.
- What is the ratio of the number of gallons of water in the rectangular basin to the cylindrical basin?\[\frac{200,000\,\, gallons}{400,000\,\, gallons}=\frac{2}{4}=\frac{1}{2}\nonumber \]
- What is the ratio of the number of gallons of water in the cylindrical basin to the rectangular basin?\[\frac{400,000\,\, gallons}{200,000\,\, gallons}=\frac{4}{2}=\frac{2}{1}=2\nonumber \]
- The price of (some chemical) recently increased from $10.80/case to $13.50/case. Find the ratio of the increase in price to the original price.
The increase in price is determined by subtracting the original price from the ending price:increase in price = $13.50-$10.80 = $2.70.
The ratio of the increase in price to the original price = \(\frac{\$2.70}{\$10.80}=0.25\, \, \text{or}\, \, \frac{1}{4}\)


17.1.1: Cofactors and Helpers in Metabolic Reactions
Enzymes and Hormones Regulate Metabolism
The human body uses enzymes to facilitate many metabolic reactions. Several of these enzymes require cofactors to function; some textbooks refer to them as coenzymes . To keep things simple, we are including these coenzymes in our definition of cofactors. Cofactors can be either organic or inorganic molecules that are required by enzymes to function. Many organic cofactors are vitamins or molecules derived from vitamins. Most inorganic cofactors are minerals. Cofactors can be oxidized or reduced for the enzymes to catalyze the reactions.
Vitamin Cofactors - examples NAD and FAD
Two common cofactors that are derived from the B vitamins, niacin and riboflavin: they are nicotinamide adenine dinucleotide (NAD) and flavin adenine dinucleotide (FAD), respectively. The structure of NAD and FAD are shown below.
Figure 9.2.1 Structure of NAD 1 upside down. The atoms are circled to help orient this structure with Figure 9.2.3
Figure 9.2.2 Structure of FAD 2 . The nitrogens are circled to help orient this structure with Figure 9.2.4
Both of these cofactors can be reduced; NAD is reduced to form NADH, while FAD is reduced to form FADH 2 as shown in the 2 figures below.
Figure 9.2.3 The reduction of NAD to form NADH 3
Figure 9.2.4 The reduction of FAD 4 to FADH 2
Accordingly, without adequate niacin and riboflavin in the diet, the coenzyme forms NAD + NAD+ and FAD would not be made in the body and metabolism would come to a stop,5 which in turn can lead to serious illness.
Mineral Cofactor - example 'iron' or Fe
An example of a mineral that serves as a cofactor is Fe 2+ for proline and lysyl hydroxylases. We will discuss later in detail why vitamin C (ascorbic acid) is needed to reduce iron to Fe 2+ so that it can serve as a cofactor for proline and lysyl hydroxylases.
Figure 9.2.5 Iron (Fe 2+ ) is a cofactor for proline and lysyl hydroxylases
Hormones that support metabolic functions
|
Hormone |
Produced by | Type of Reaction | Control of Carbohydrate Metabolism | Control of Fat Metabolism | Control of Protein Metabolism | |
|
Cortisol |
Adrenal glands | Catabolic | Stimulates gluconeogenesis | In acute states, no effect. Chronically elevated levels lead to lipogenesis | Stimulates protein degradation | |
| Epinephrine | Adrenal glands | Catabolic | Stimulates glycogenolysis | Stimulates lipolysis | May stimulate protein degradation | |
| Glucagon | Pancreas | Catabolic | Stimulates glycogenolysis | Stimulates lipolysis | Stimulates protein degradation | |
| Insulin |
|
Anabolic | Stimulates glycogen synthesis | Stimulates lipogenesis | Stimulates protein synthesis |
Table 9.2 Major Hormones with influence on metabolism (modified from5)
Check your knowledge:
True or False: Enzymes catalyze metabolic reactions with the aid of cofactors, especially forms of niacin and riboflavin, as well as hormones such as insulin, glucagon, epinephrine, and cortisol.
True or False: Insulin and glucagon are produced in the pancreas, cortisol and epinephrine are produced in the adrenal glands.
References & Links
- en.Wikipedia.org/wiki/File:NAD%2B_phys.svg
- en.Wikipedia.org/wiki/File:Fl...nucleotide.png
- en.Wikipedia.org/wiki/File:NA..._reduction.svg
- en.Wikipedia.org/wiki/File:FA...equlibrium.png
- Blake, J. S., Munoz, K. D., & Volpe, S. (2019). Nutrition: From Science to You (4th ed.). Pearson.


3.1: Active Listening
Active listening is a way of listening and responding to another person so that the message is fully understood. The following are several techniques that you can use to demonstrate active listening. The techniques you use will vary depending on the situation. For example, active listening during a lecture will require different techniques than active listening about a personnel matter at the job.
Concentrate
Eliminate distractions.  Try to put personal problems aside. Limit engagement in other activities such as texting or working on other assignments. If you are having difficulty concentrating, use techniques to keep your mind from wandering. This may include taking very brief notes or jotting down questions you might want to ask at the appropriate time.
Empathize
Put yourself inside the speaker’s thoughts and feelings to better understand what he or she is saying to you. Suspend your own judgment and position until you clearly understand the other person’s perspective.
Listen for feelings
Try to “listen between the lines” to understand the attitudes, needs, and motives behind the words. Changes in volume and tone, as well as non-verbal clues such as facial expressions and gestures, can help you determine how the speaker is feeling.
Connect
Use “listener-friendly” body language: make eye contact with the speaker or focus on the audio or visual presentation at hand. Try to connect the information you are hearing with what you may have previously learned or already know. Pay attention to any visuals that may accompany the audio, such as, an instructor writing on a board or asking you to look at a visual in your textbook or online while they continue speaking.
Validate
Even if you don’t agree with what the speaker is saying, it is important that the person knows you are listening and that you understand what they have said. Use nods and “uh-huhs” and respectful comments that show you have heard what was said.
Paraphrase
When the speaker has finished talking, repeat in your own words what the speaker said so they know they have been understood.
Clarify
Ask questions to get more information, especially if you’re not clear on what was said. It is important to take your cues from the presenter on when to ask questions. While some instructors may ask you to interrupt and ask questions at any time, others may ask you to hold questions until the appropriate time.
Participate
Participate in discussions and respond to questions.
Now complete the Learning Task Self-Test.


9.1: Simplify Rational Expressions
A rational expression is written as a quotient of polynomials
\[\dfrac{P(x)}{Q(x)} \nonumber \]
where \(P(x)\) and \(Q(x)\) are polynomials in one variable \(x\).
To simplify a rational expression, factor both the numerator and the denominator, and remove common factors from both the numerator and the denominator. A simplified rational expression has only one division, and a single numerator and denominator. If the expressions cannot be factored, then the rational expression cannot be simplified.
Simplify the rational expressions:
- \(\dfrac{x^2 + 2x − 3}{x^2 + 4x + 3}\)
- \(\dfrac{(x^2 + 1)^2 (−2) + (2x)(2)(x^2 + 1)(2x)}{(x^2 + 1)^4}\)
- \(\dfrac{(x^2 + 1) \frac{1}{2} (x^{−\frac{1}{2}}) − (2x)(x^{\frac{1}{2}})}{(x^2 + 1)^2}\)
Solution
- \(\begin{array} &&\dfrac{x^2 + 2x − 3}{x^2 + 4x + 3} &\text{Example problem} \\ &\dfrac{(x + 3)(x − 1)}{(x + 3)(x + 1)} &\text{Factor both numerator and denominator.} \\ &\dfrac{\cancel{(x + 3)}(x − 1)}{\cancel{(x + 3)}(x + 1)} &\text{Remove common factors, because \(\dfrac{x + 3}{x + 3} = 1\)} \\ &\dfrac{x − 1}{x + 1} &\text{Final answer} \end{array}\)
- \(\begin{array} &&\dfrac{(x^2 + 1)^2 (−2) + (2x)(2)(x^2 + 1)(2x)}{(x^2 + 1)^4} &\text{Example problem} \\ &\dfrac{2(x^2 + 1)[(x^2 + 1)(−1) + (2x)(2x)]}{(x^2 + 1)^4} &\text{Factor out 2(x^2 + 1)} \\ &\dfrac{2 \cancel{(x^2 + 1)}[(x^2 + 1)(−1) + (2x)(2x)]}{\cancel{(x^2+1)}(x^2 + 1)^3} &\text{Remove common factors, because \(\dfrac{x^2 + 1}{x^2 + 1} = 1\)} \\ &\dfrac{2[−x^2 − 1 + 4x^2]}{(x^2 + 1)^3} &\text{Simplify by multiplying and combining like terms} \\ &\dfrac{2(3x^2 − 1)}{(x^2 + 1)^3} &\text{Final answer} \end{array}\)
- \(\begin{array} &&\dfrac{(x^2 + 1) \frac{1}{2} (x^{−\frac{1}{2}}) − (2x)(x^{\frac{1}{2}})}{(x^2 + 1)^2} &\text{Example problem} \\ &\dfrac{\frac{(x^2+1)}{2x^{\frac{1}{2}}} − (2x)(x^{ \frac{1}{2} })}{(x^2 + 1)^2} &\text{Work with the negative exponent in the first term of the numerator by moving the factor to the denominator of the first term, next to the \(2\).} \\ &\dfrac{(x^2 + 1) − (2x)(x^{\frac{1}{2}} )2(x^{\frac{1}{2}} )}{\dfrac{2x^{\frac{1}{2}} }{(x^2 + 1)^2}} &\text{Common denominator} \\ &\dfrac{x^2 + 1 − 4x^2}{(2x^{\frac{1}{2}} )(x^2 + 1)^2} &\text{Simplify by multiplying and combining like terms} \\ &\dfrac{−3x^2 + 1}{(2x^{\frac{1}{2}} )(x^2 + 1)^2} &\text{Final answer} \end{array}\)
Simplify the rational expressions:
- \(\dfrac{2x^2 + 3x − 2}{2x^2 + 5x − 3}\)
- \(\dfrac{(t^2 + 4)(2t − 4) − (t^2 − 4t + 4)(2t)}{(t^2 + 4)^2}\)
- \(\dfrac{(2)(x − 4)(x^2 + 4x + 4)}{(x + 2)(x^2 − 16)}\)
- \(\dfrac{12x^2 + 19x − 21}{12x^2 + 38x − 40}\)


12.6.5: Procedure- Assisting with Emptying of the Urinary Drainage Bag
Drainage bags on urinary collection systems, such as those from indwelling catheters or condom catheters need to be emptied on a regular basis. Home Health Aides/Personal Care Aides should frequently check that the tubing on catheter systems is not twisted or kinked. Home Health Aides/Personal Care Aides should always ensure that drainage bags are below bladder level. Never hang them from bed rails as the bag will move when bed rails are raised or lowered.
- Explain the procedure to the patient.
- Wash your hands and don gloves.
- Assemble equipment needed (measuring pitcher or graduate cylinder).
- Place the measuring pitcher or graduate cylinder below the drainage bag on a paper towel.
- Release the drain from the holder.
- Open the clamp on the drainage bag without allowing the drain to touch the measuring cylinder.
- Allow contents to pour into measuring container.
- Take care to prevent urine from splashing onto your face or clothing.
- Clamp the drainage bag and clean the end of the drain with an alcohol wipe.
- Position the clamp back into its holder.
- Measure the amount of urine at eye level while it is sitting on a flat surface. Record this as output.
- Dispose of urine in the toilet. Flush. Clean the measuring container and store.
- Discard gloves and wash your hands.
- Document procedure, output, and any observed changes in behavior or condition. Report observations about changes in urine color, odor, amount, or characteristics, such as cloudiness, mucus or blood present. Note the condition of the drainage bag and catheter tubing. Report concerns or the need to replace catheter tubing or the drainage bag to a supervisor.


14: Antimicrobial Drugs
In nature, some microbes produce substances that inhibit or kill other microbes that might otherwise compete for the same resources. Humans have successfully exploited these abilities, using microbes to mass-produce substances that can be used as antimicrobial drugs. Since their discovery, antimicrobial drugs have saved countless lives, and they remain an essential tool for treating and controlling infectious disease. But their widespread and often unnecessary use has had an unintended side effect: the rise of multidrug-resistant microbial strains. In this chapter, we will discuss how antimicrobial drugs work, why microbes develop resistance, and what health professionals can do to encourage responsible use of antimicrobials.
-
- 14.1: Discovering Antimicrobial Drugs
- Antimicrobial drugs produced by purposeful fermentation and/or contained in plants have been used as traditional medicines in many cultures for millennia. The purposeful and systematic search for a chemical “magic bullet” that specifically target infectious microbes was initiated by Paul Ehrlich in the early 20th century. The discovery of the natural antibiotic, penicillin, by Alexander Fleming in 1928 started the modern age of antimicrobial discovery and research.
-
- 14.2: Antibacterial Drugs
- Antimicrobial drugs can be bacteriostatic or bactericidal, and these characteristics are important considerations when selecting the most appropriate drug. The use of narrow-spectrum antimicrobial drugs is preferred in many cases to avoid superinfection and the development of antimicrobial resistance. Broad-spectrum antimicrobial use is warranted for serious systemic infections when there is no time to determine the causative agent or when narrow-spectrum antimicrobials fail.
-
- 14.3: Drugs Targeting Other Microorganisms
- Antibacterial compounds exhibit selective toxicity, largely due to differences between prokaryotic and eukaryotic cell structure. Cell wall synthesis inhibitors, including the β-lactams, the glycopeptides, and bacitracin, interfere with peptidoglycan synthesis, making bacterial cells more prone to osmotic lysis. There are a variety of broad-spectrum, bacterial protein synthesis inhibitors that selectively target the prokaryotic 70S ribosome, including those that bind to the 30S and 50S subunits.
-
- 14.5: Testing the Effectiveness of Antimicrobials
- Antimicrobial resistance is on the rise and is the result of selection of drug-resistant strains in clinical environments, the overuse and misuse of antibacterials, the use of subtherapeutic doses of antibacterial drugs, and poor patient compliance with antibacterial drug therapies. Drug resistance genes are often carried on plasmids or in transposons that can undergo vertical transfer easily and between microbes through horizontal gene transfer.
-
- 14.6: The Emergence of Drug Resistance
- The Kirby-Bauer disk diffusion test helps determine the susceptibility of a microorganism to various antimicrobial drugs. However, the zones of inhibition measured must be correlated to known standards to determine susceptibility and resistance, and do not provide information on bactericidal versus bacteriostatic activity, or allow for direct comparison of drug potencies. Antibiograms are useful for monitoring local trends in antimicrobial resistance/susceptibility.
Footnotes
- 1 “Treatment of War Wounds: A Historical Review.” Clinical Orthopaedics and Related Research 467 no. 8 (2009):2168–2191.
Thumbnail: Staphylococcus aureus - Antibiotics Test plate. (Public Domain; CDC / Provider: Don Stalons).


2.6: The geometry of matrix transformations
\(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\twovec}[2]{\begin{pmatrix} #1 \\ #2 \end{pmatrix} } \) \(\newcommand{\threevec}[3]{\begin{pmatrix} #1 \\ #2 \\ #3 \end{pmatrix} } \) \(\newcommand{\fourvec}[4]{\begin{pmatrix} #1 \\ #2 \\ #3 \\ #4 \end{pmatrix} } \) \(\newcommand{\fivevec}[5]{\begin{pmatrix} #1 \\ #2 \\ #3 \\ #4 \\ #5 \end{pmatrix} } \)Matrix transformations, which we explored in the last section, allow us to describe certain functions \(T:\mathbb R^n\to\mathbb R^m\text{.}\) In this section, we will demonstrate how matrix transformations provide a convenient way to describe geometric operations, such as rotations, reflections, and scalings. We will then explore how matrix transformations are used in computer animation.
Preview Activity 2.6.1.
Suppose that we wish to describe the geometric operation that reflects 2-dimensional vectors in the horiztonal axis. For instance, Figure 2.6.1 illustrates how a vector \(\mathbf x\) is reflected into the vector \(T(\mathbf x)\text{.}\)
- If \(\mathbf x = \twovec{2}{4}\text{,}\) what is the vector \(T(\mathbf x)\text{?}\) Sketch the vectors \(\mathbf x\) and \(T(\mathbf x)\text{.}\)
- More generally, if \(\mathbf x=\twovec{x}{y}\text{,}\) what is \(T(\mathbf x)\text{?}\)
- Find the vectors \(T\left(\twovec{1}{0}\right)\) and \(T\left(\twovec{0}{1}\right)\text{.}\)
- Use your results to write the matrix \(A\) so that \(T(\mathbf x) = A\mathbf x\text{.}\) Then verify that \(T\left(\twovec{x}{y}\right)\) agrees with what you found in part b.
- Describe the transformation that results from composing \(T\) with itself; that is, what is the transformation \(T\circ T\text{?}\) Explain how matrix multiplication can be used to justify your response.
The geometry of \(2\times2\) matrix transformations
The preview activity demonstrates how the matrix \(\left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \\ \end{array}\right]\) defines a matrix transformation that has the effect of reflecting 2-dimensional vectors in the horizontal axis. The following activity shows, more generally, that matrix transformations can perform a variety of important geometric operations.
Activity 2.6.2.
The diagram below demonstrates the effect of a matrix transformation on the plane. You may modify the matrix \(A\) defining the matrix transformation \(T\) through the sliders at the top. You may also move the red vector \({\mathbf x}\) on the left, by clicking in the head of the vector, and observe \(T({\mathbf x})\) on the right.
For the following matrices \(A\) given below, use the diagram to study the effect of the corresponding matrix transformation \(T(\mathbf x) = A\mathbf x\text{.}\) For each transformation, describe the geometric effect of the transformation on the plane.
- The matrix \(A=\left[\begin{array}{rr} 2 & 0 \\ 0 & 1 \\ \end{array}\right]\text{.}\)
- The matrix \(A=\left[\begin{array}{rr} 2 & 0 \\ 0 & 2 \\ \end{array}\right]\text{.}\)
- The matrix \(A=\left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \\ \end{array}\right]\text{.}\)
- The matrix \(A=\left[\begin{array}{rr} 1 & 1 \\ 0 & 1 \\ \end{array}\right]\text{.}\)
- The matrix \(A=\left[\begin{array}{rr} -1 & 0 \\ 0 & 1 \\ \end{array}\right]\text{.}\)
- The matrix \(A=\left[\begin{array}{rr} 1 & 0 \\ 0 & 0 \\ \end{array}\right]\text{.}\)
- The matrix \(A=\left[\begin{array}{rr} 1 & 0 \\ 0 & 1 \\ \end{array}\right]\text{.}\)
- The matrix \(A=\left[\begin{array}{rr} 1 & -1 \\ -2 & 2 \\ \end{array}\right]\text{.}\)
The previous activity presented some examples in which matrix transformations perform interesting geometric actions, such as rotations, scalings, and reflections. Let's turn this question around: Suppose we have a specific geometric action that we would like to perform. Can we find a matrix \(A\) that represents this action through the matrix transformation \(T(\mathbf x) = A\mathbf x\text{?}\)
The linearity of matrix-vector multiplication Proposition 2.2.3 provides the key to answering this question. Remember that if \(A\) is a matrix, \(\mathbf v\) and \(\mathbf w\) vectors, and \(c\) a scalar, then
This means that a matrix transformation \(T(\mathbf x) = A\mathbf x\) satisfies the corresponding linearity property:
Linearity of Matrix Transformations.
It turns out that, if \(T:\mathbb R^n\to\mathbb R^m\) satisfies these two linearity properties, then we can find a matrix \(A\) such that \(T(\mathbf x) = A\mathbf x\text{.}\) In fact, Proposition 2.5.4 tells us how to form \(A\text{;}\) we simply write
We will now check that \(T(\mathbf x) = A\mathbf x\) using the linearity of \(T\text{:}\)
The result is the following proposition.
The function \(T:\mathbb R^n\to\mathbb R^m\) is a matrix transformation where \(T(\mathbf x) = A\mathbf x\) for some \(m\times n\) matrix \(A\) if and only if
In this case, \(A\) is the matrix whose columns are \(T(\mathbf e_j)\text{;}\) that is,
We will put this proposition to use in the following example by finding the matrix whose matrix transformation performs a specific geometric operation.
In this example, we will find the matrix defining a matrix transformation that performs a \(45^\circ\) counterclockwise rotation.
We first need to know that this geometric operation can be represented by a matrix transformation. To begin, we will define the function \(T:\mathbb R^2\to\mathbb R^2\) where \(T(\mathbf x)\) is obtained by rotating \(\mathbf x\) counterclockwise by \(45^\circ\text{,}\) as shown in Figure 2.6.4.
We need to check that \(T\) is a matrix transformation; by Proposition 2.6.2, this means that we should make sure that
The next two figures illustrate these properties. For instance, Figure 2.6.5 shows that relationship between \(T(\mathbf v)\) and \(T(c\mathbf v)\) when \(c\) is a scalar. We easily see that \(T(c\mathbf v)\) is a scalar multiple of \(T(\mathbf v)\) and hence that \(T(c\mathbf v) = cT(\mathbf v)\text{.}\)
Similarly, Figure 2.6.6 shows the relationship between \(T(\mathbf v+\mathbf w)\text{,}\) \(T(\mathbf v)\text{,}\) and \(T(\mathbf w)\text{.}\) In this way, we see that \(T(\mathbf v+\mathbf w) = T(\mathbf v) + T(\mathbf w)\text{.}\)
This shows that the function \(T\text{,}\) which rotates vectors by \(45^\circ\) is a matrix transformation. We may therefore write it as \(T(\mathbf x) = A\mathbf x\) where \(A\) is the \(2\times2\) matrix \(A=\left[\begin{array}{rr} T(\mathbf e_1) & T(\mathbf e_2) \end{array}\right]\text{.}\) The columns of this matrix, \(T(\mathbf e_1)\) and \(T(\mathbf e_2)\text{,}\) are shown in Figure 2.6.7.
To find the components of these vectors, notice that they form an isosceles right triangle, as shown in Figure 2.6.8. Since the length of \(\mathbf e_1\) is 1, the length of \(T(\mathbf e_1)\text{,}\) the hypotenuse of the triangle, is 1.
This leads to
Hence, the matrix \(A\) is
You may wish to check this using the interactive diagram in the previous activity using the approximation \(1/\sqrt{2} \approx 0.7\text{.}\)
In this example, we found that the desired geometric operation, a rotation in the plane, was in fact a matrix transformation \(T\) by checking that
In general, the same kind of thinking applies to show that rotations, reflections, and scalings are matrix transformations so we will not bother with that step in the future.
Activity 2.6.3.
In this activity, we seek to describe various matrix transformations by finding the matrix that gives the desired transformation. All of the transformations that we study here have the form \(T:\mathbb R^2\to\mathbb R^2\text{.}\)
- Find the matrix of the transformation that has no effect on vectors; that is, \(T(\mathbf x) = \mathbf x\text{.}\) We call this matrix the identity and denote it by \(I\text{.}\)
- Find the matrix of the transformation that reflects vectors in \(\mathbb R^2\) over the line \(y=x\text{.}\)
- What is the result of composing the reflection you found in the previous part with itself; that is, what is the effect of reflecting in the line \(y=x\) and then reflecting in this line again. Provide a geometric explanation for your result as well as an algebraic one obtained by multiplying matrices.
- Find the matrix that rotates vectors counterclockwise in the plane by \(90^\circ\text{.}\)
- Compare the result of rotating by \(90^\circ\) and then reflecting in the line \(y=x\) to the result of first reflecting in \(y=x\) and then rotating \(90^\circ\text{.}\)
- Find the matrix that results from composing a \(90^\circ\) rotation with itself. Explain the geometric meaning of this operation.
- Find the matrix that results from composing a \(90^\circ\) rotation with itself four times; that is, if \(T\) is the matrix transformation that rotates vectors by \(90^\circ\text{,}\) find the matrix for \(T\circ T\circ T \circ T\text{.}\) Explain why your result makes sense geometrically.
-
Explain why the matrix that rotates vectors counterclockwise by an angle \(\theta\) is
\begin{equation*} \left[\begin{array}{rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{array}\right]\text{.} \end{equation*}
In the first part of this activity, we encountered the identity matrix, which, as an \(n\times n\) matrix, has the form
The matrix transformation \(T(\mathbf x) = I\mathbf x\) leaves vectors unchanged; that is, \(T(\mathbf x) = \mathbf x\) so that \(I\mathbf x = \mathbf x\text{.}\) Notice that the columns of \(I\) are simply the vectors \(\mathbf e_j\text{.}\)
Matrix transformations and computer animation
Linear algebra plays a significant role in computer animation. We will now illustrate how matrix transformations and some of the ideas we have developed in this section are used by computer animators to create the illusion of motion in their characters.
Figure 2.6.9 shows a test character used by Pixar animators. On the left is the original definition of the character; on the right, we see that the character has been moved into a different pose. To make it appear that the character is moving, animators create a sequence of frames in which the character's pose is modified slightly from one frame to the next. Matrix transformations play an important role in doing this.
For instance, Figure 2.6.10 shows the character Remy from Pixar's Ratatouille . Clearly, a lot goes into transforming the model on the left into the engaging character on the right, such as the addition of fur and eyes. We will focus only on the motion of the character.
Of course, realistic characters will be drawn in three-dimensions. To keep things a little more simple, however, we will look at this two-dimensional character and devise matrix transformations that move them into different poses.
Of course, the first thing we may wish to do is simply move them to a different position in the plane, such as that shown in Figure 2.6.11. Motions like this are called translations .
This presents a problem because a matrix transformation \(T:\mathbb R^2\to\mathbb R^2\) has the property that \(T(\zerovec) = \zerovec\text{.}\) This means that a matrix transformation cannot move the origin of the coordinate plane. To address this restriction, animators use homogeneous coordinates , which are formed by placing the two-dimensional coordinate plane inside \(\mathbb R^3\) as the plane \(z=1\text{.}\) This is shown in Figure 2.6.12.
Therefore, rather than describing points in the plane as vectors \(\twovec{x}{y}\text{,}\) we describe them as three-dimensional vectors \(\threevec{x}{y}{1}\text{.}\) As we see in the next activity, this allows us to translate our character in the plane.
Activity 2.6.4.
In this activity, we will use homogeneous coordinates and matrix transformations to move our character into a variety of poses.
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Since we regard our character as living in \(\mathbb R^3\text{,}\) we will consider matrix transformations defined by matrices
\begin{equation*} \left[\begin{array}{rrr} a & b & c \\ d & e & f \\ 0 & 0 & 1 \\ \end{array}\right]\text{.} \end{equation*}
Verify that such a matrix transformation transforms points in the plane \(z=1\) into other points in this plane; that is, verify that
\begin{equation*} \left[\begin{array}{rrr} a & b & c \\ d & e & f \\ 0 & 0 & 1 \\ \end{array}\right] \threevec{x}{y}{1} = \threevec{x'}{y'}{1}\text{.} \end{equation*}Express the coordinates of the resulting point \(x'\) and \(y'\) in terms of the coordinates of the original point \(x\) and \(y\text{.}\)
The diagram below allows you to create matrix transformations of this form to move our character into different poses. You may use it to help address the following questions. - Find the matrix transformation that translates our character to a new position in the plane, as shown in Figure 2.6.13
- As originally drawn, our character is waving with one of their hands. In one of the movie's scenes, we would like her to wave with their other hand, as shown in Figure 2.6.14. Find the matrix transformation that moves them into this pose.
- Later, our chracter performs a cartwheel by moving through the sequence of poses shown in Figure 2.6.15. Find the matrix transformations that create these poses.
- Next, we would like to find the transformations that zoom in on our character's face, as shown in Figure 2.6.16. To do this, you should think about composing matrix transformations. This can be accomplished in the diagram by using the Compose button, which makes the current pose, displayed on the right, the new beginning pose, displayed on the left. What is the matrix transformation that moves the character from the original pose, shown in the upper left, to the final pose, shown in the lower right?
- We would also like to create our character's shadow, shown in the sequence of poses in Figure 2.6.17. Find the sequence of matrix transformations that achieves this. In particular, find the matrix transformation that take our character from their original pose to their shadow in the lower right.
- Write a final scene to the movie and describe how to construct a sequence of matrix transformations that create your scene.
Summary
This section explored how geometric operations, such as rotations, reflections, and scalings, are performed by matrix transformations.
- A matrix of the form \(\left[\begin{array}{rr} a & 0 \\ 0 & b \\ \end{array}\right]\) represents a horizontal scaling by a factor \(a\) and a vertical scaling by \(b\text{.}\)
- A matrix of the form \(\left[\begin{array}{rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{array}\right]\) defines a rotation by an angle \(\theta\text{.}\)
- Composing geometric operations corresponds to matrix multiplication.
- Computer animators use matrix transformations to create the illusion of motion. Homogeneous coordinates are used so that translations can be realized as matrix transformations.
Exercises 2.6.4Exercises
For each of the following geometric operations in the plane, find a \(2\times 2\) matrix that defines the matrix transformation performing the operation.
- Rotates vectors by \(180^\circ\text{.}\)
- Reflects vectors in the vertical axis.
- Reflects vectors in the line \(y=-x\text{.}\)
- Rotates vectors counterclockwise by \(60^\circ\text{.}\)
- First rotates vectors counterclockwise by \(60^\circ\) and then reflects in the line \(y=x\text{.}\)
This exercise investigates the composition of reflections in the plane.
- Find the result of first reflecting in the line \(y=0\) and then \(y=x\text{.}\) What familiar operation is the cumulative effect of this composition?
- What happens if you compose the operations in the opposite order; that is, what happens if you first reflect in \(y=x\) and then \(y=0\text{?}\) What familiar operation results?
- What familiar geometric operation results if you first reflect in the line \(y=x\) and then \(y=-x\text{?}\)
- What familiar geometric operation results if you first rotate by \(90^\circ\) and then reflect in the line \(y=x\text{?}\)
It is a general fact that the composition of two reflections results in a rotation through twice the angle from the first line of reflection to the second. We will investigate this more generally in Exercise <IP_ADDRESS>
Shown below in Figure 2.6.18 are the vectors \(\mathbf e_1\text{,}\) \(\mathbf e_2\text{,}\) and \(\mathbf e_3\) in \(\mathbb R^3\text{.}\)
- Imagine that the thumb of your right hand points in the direction of \(\mathbf e_1\text{.}\) A positive rotation about the \(x\) axis corresponds to a rotation in the direction in which your fingers point. Find the matrix definining the matrix transformation \(T\) that rotates vectors by \(90^\circ\) around the \(x\)-axis.
- In the same way, find the matrix that rotates vectors by \(90^\circ\) around the \(y\)-axis.
- Find the matrix that rotates vectors by \(90^\circ\) around the \(z\)-axis.
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What is the cumulative effect of rotating by \(90^\circ\) about the \(x\)-axis, followed by a \(90^\circ\) rotation about the \(y\)-axis, followed by a \(-90^\circ\) rotation about the \(x\)-axis.
We have seen how a matrix transformation can perform a geometric operation; now we would like to find a matrix transformation that undoes that operation.
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Suppose that \(T:\mathbb R^2\to\mathbb R^2\) is the matrix transformation that rotates vectors by \(90^\circ\text{.}\) Find a matrix transformation \(S:\mathbb R^2\to\mathbb R^2\) that undoes the rotation; that is, \(S\) takes \(T(\mathbf x)\) back into \(\mathbf x\) so that \(S\circ T(\mathbf x) = \mathbf x\text{.}\) Think geometrically about what the transformation \(S\) should be and then verify it algebraically.
We say that \(S\) is the inverse of \(T\) and we will write it as \(T^{-1}\text{.}\)
- Verify algebraically that the reflection \(R:\mathbb R^2\to\mathbb R^2\) across the line \(y=x\) is its own inverse; that is, \(R^{-1} = R\text{.}\)
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The matrix transformation \(T:\mathbb R^2\to\mathbb R^2\) defined by the matrix
\begin{equation*} A= \left[\begin{array}{rr} 1& 1 \\ 0 & 1\\ \end{array}\right] \end{equation*}
is called a shear . Find the inverse of \(T\text{.}\)
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Describe the geometric effect of the matrix transformation defined by
\begin{equation*} A=\left[\begin{array}{rr} \frac12 & 0 \\ 0 & 3\\ \end{array}\right] \end{equation*}
and then find its inverse.
We have seen that the matrix
performs a rotation through an angle \(\theta\) about the origin. Suppose instead that we would like to rotate by \(90^\circ\) about the point \((1,2)\text{.}\) Using homogeneous coordinates, we will develop a matrix that performs this operation.
Our strategy is to
- begin with a vector whose tail is at the point \((1,2)\text{,}\)
- translate the vector so that its tail is at the origin,
- rotate by \(90^\circ\text{,}\) and
- translate the vector so that its tail is back at \((1,2)\text{.}\)
This is shown in Figure 2.6.19.
Remember that, when working with homogeneous coordinates, we consider matrices of the form
- The first operation is a translation by \((-1,-2)\text{.}\) Find the matrix that performs this translation.
- The second operation is a \(90^\circ\) rotation about the origin. Find the matrix that performs this rotation.
- The third operation is a translation by \((1,2)\text{.}\) Find the matrix that performs this translation.
- Use these matrices to find the matrix that performs a \(90^\circ\) rotation about \((1,2)\text{.}\)
- Use your matrix to determine where the point \((-10, 5)\) ends up if rotated by \(90^\circ\) about the \((1,2)\text{.}\)
This exercise concerns matrix transformations called projections .
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Consider the matrix transformation \(T:\mathbb R^2\to\mathbb R^2\) that assigns to a vector \(\mathbf x\) the closest vector on horizontal axis as illustrated in Figure 2.6.20. This transformation is called the projection onto the horizontal axis. You may imagine \(T(\mathbf x)\) as the shadow cast by \(\mathbf x\) from a flashlight far up on the positive \(y\)-axis.
Find the matrix that defines this matrix transformation \(T\text{.}\)
- Find the matrix that defines projection on the vertical axis.
- What is the result of composing the projection onto the horizontal axis with the projection onto the vertical axis?
- Find the matrix that defines projection onto the line \(y=x\text{.}\)
This exericse concerns the matrix transformations defined by matrices of the form
Let's begin by looking at two special types of these matrices.
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First, consider the matrix where \(a = 2\) and \(b=0\) so that
\begin{equation*} A = \left[\begin{array}{rr} 2 & 0 \\ 0 & 2 \\ \end{array}\right]\text{.} \end{equation*}
Describe the geometric effect of this matrix. More generally, suppose we have
\begin{equation*} A = \left[\begin{array}{rr} r & 0 \\ 0 & r \\ \end{array}\right]\text{,} \end{equation*}where \(r\) is a positive number. What is the geometric effort of \(A\) on vectors in the plane?
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Suppose now that \(a = 0\) and \(b = 1\) so that
\begin{equation*} A = \left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \\ \end{array}\right]\text{.} \end{equation*}
What is the geometric effect of \(A\) on vectors in the plane? More generally, suppose we have
\begin{equation*} A = \left[\begin{array}{rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{array}\right]\text{.} \end{equation*}What is the geometric effect of \(A\) on vectors in the plane?
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In general, the composition of matrix transformation depends on the order in which we compose them. For these transformations, however, it is not the case. Check this by verifying that
\begin{equation*} \left[\begin{array}{rr} r & 0 \\ 0 & r \\ \end{array}\right] \left[\begin{array}{rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{array}\right] = \left[\begin{array}{rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{array}\right] \left[\begin{array}{rr} r & 0 \\ 0 & r \\ \end{array}\right]\text{.} \end{equation*}
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Let's now look at the general case where
\begin{equation*} A = \left[\begin{array}{rr} a & -b \\ b & a \\ \end{array}\right]\text{.} \end{equation*}
We will draw the vector \(\twovec{a}{b}\) in the plane and express it using polar coordinates \(r\) and \(\theta\) as shown in Figure 2.6.21.
We then have
\begin{equation*} \twovec{a}{b} = \twovec{r\cos\theta}{r\sin\theta}\text{.} \end{equation*}Show that the matrix
\begin{equation*} \left[\begin{array}{rr} a & -b \\ b & a \\ \end{array}\right] = \left[\begin{array}{rr} r & 0 \\ 0 & r \\ \end{array}\right] \left[\begin{array}{rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{array}\right]\text{.} \end{equation*} -
Using this description, describe the geometric effect on vectors in the plane of the matrix transformation defined by
\begin{equation*} A= \left[\begin{array}{rr} a & -b \\ b & a \\ \end{array}\right]\text{.} \end{equation*}
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Suppose we have a matrix transformation \(T\) defined by a matrix \(A\) and another transformation \(S\) defined by \(B\) where
\begin{equation*} A= \left[\begin{array}{rr} a & -b \\ b & a \\ \end{array}\right], B= \left[\begin{array}{rr} c & -d \\ d & c \\ \end{array}\right]\text{.} \end{equation*}
Describe the geometric effect of the composition \(S\circ T\) in terms of the \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) and \(d\text{.}\)
The matrices of this form give a model for the complex numbers and will play an important role in Section 4.4 .
We saw earlier that the rotation in the plane through an angle \(\theta\) is given by the matrix:
We would like to find a similar expression for the matrix that represents the reflection in \(L_\theta\text{,}\) the line passing through the origin and making an angle of \(\theta\) with the positive \(x\)-axis, as shown in Figure 2.6.22.
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To do this, notice that this reflection can be obtained by composing three separate transformations as shown in Figure 2.6.23. Beginning with the vector \(\mathbf x\text{,}\) we apply the transformation \(R\) to rotate by \(-\theta\) and obtain \(R(\mathbf x)\text{.}\) Next, we apply \(S\text{,}\) a reflection in the horizontal axis, followed by \(T\text{,}\) a rotation by \(\theta\text{.}\) We see that \(T(S(R(\mathbf x)))\) is the same as the reflection of \(\mathbf x\) in the original line \(L_\theta\text{.}\)
Using this decomposition, show that the reflection in the line \(L_\theta\) is described by the matrix
\begin{equation*} \left[\begin{array}{rr} \cos(2\theta) & \sin(2\theta) \\ \sin(2\theta) & -\cos(2\theta) \\ \end{array}\right]\text{.} \end{equation*}You will need to remember the trigonometric identities:
\begin{equation*} \begin{aligned} \cos(2\theta) & {}={} \cos^2\theta - \sin^2\theta \\ \sin(2\theta) & {}={} 2\sin\theta\cos\theta \\ \end{aligned}\text{.} \end{equation*} - Now that we have a matrix that describes the reflection in the line \(L_\theta\text{,}\) show that the composition of the reflection in the horizontal axis followed by the reflection in \(L_\theta\) is a counterclockwise rotation by an angle \(2\theta\text{.}\) We saw some examples of this earlier in Exercise <IP_ADDRESS>.


15.23C: Structure of the Lymphatic System
- Describe the structure and function of the lymphatic system
The lymphatic system is a collection of structures and vessels that drains lymph from blood and has several other functions. It is a circulatory system for lymph fluid and the site of many key immune system functions.
Lymphatic Vessels
The lymphatic vessels are the lymphatic system equivalent of the blood vessels of the circulatory system and drain fluid from the circulatory system. The network of lymph vessels consists of the initial collectors of lymph fluid, which are small, valveless vessels, and goes on to form the precollector vessels, which have rudimentary valves that are not fully functional. These structures then form increasingly larger lymphatic vessels which form colaterals and have lymph-angions (lymph hearts). The larger lymph vessels contain valves that prevent the backflow of lymph.
The lymphatic system is an active pumping system driven by segments that have a function similar to peristalsis. They lack a central pump (like the heart in the cardio vascular system), so smooth muscle tissue contracts to move lymph along through the vessels. Skeletal muscle contractions also move lymph through the vessels. The lymphatic vessels make their way to the lymph nodes, and from there the vessels form into trunks. In general, the lymph vessels bring lymph fluid toward the heart and above it to the subclavian veins, which enable lymph fluid to re-enter the circulatory system through the vena cava.
Lymphatic Tissues and Organs
Lymphoid tissue is found in many organs including the lymph nodes, as well as in the lymphoid follicles in the pharynx such as the tonsils. Lymph nodes are found primarily in the armpits, groin, chest, neck, and abdomen. Lymphoid tissues contain lymphocytes (a type of highly differentiated white blood cell), but they also contain other types of cells for structural and functional support, such as the dendritic cells, which play a key role in the immune system. The system also includes all the structures dedicated to the circulation and production of lymphocytes, including the spleen, thymus, and bone marrow.
Key Points
- The lymphatic system is a circulatory system that drains fluid from the blood vessels.
- Lymph vessels are the site of fluid drainage and pump lymph fluid using smooth muscle and skeletal muscle action. The larger vessels contain valves to prevent backflow and pump towards the heart to return lymph fluid to the bloodstream by the subclavian veins.
- A lymph node is an organized collection of lymphoid tissue through which the lymph passes on its way to returning to the blood. Lymph nodes are located at intervals along the lymphatic system.
- Lymphoid tissue contains lymphocytes and other specialized cells and tissues that have immune system functions.
Key Terms
- lymph node : Small oval bodies of the lymphatic system, distributed along the lymphatic vessels clustered in the armpits, groin, neck, chest, and abdomen.


16.3: Chronic Hunger and the Civic Participant
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A different approach to governing hunger can be seen in Brazil. Although the country was for a long time a net exporter of agricultural products, it also had huge numbers of undernourished people living within its borders. This reaffirms the point that, in and of themselves, food surpluses do not prevent hunger – even at state level. So, when the left-wing Workers’ Party was elected to power in 2003, their leader Luiz Inácio Lula da Silva made the Zero Hunger programme a cornerstone of his government’s social policy. He declared in his inauguration speech: ‘We are going to create appropriate conditions for all people in our country to have three decent meals a day, every day, without having to depend on donations from anybody’ (cited in da Silva et al. 2011, 9).
This commitment came out of the country’s re-democratisation process in the 1990s, when civil society began to exert a greater influence in national politics after two decades of oppressive military dictatorship. The Council on Food and Nutritional Security, which was supported by Lula, was a particularly important institution in this respect. Composed of 54 representatives, twothirds from civil society and one-third from federal government, the Council drove forward a number of policies, including increased funding for school meals and support for family farmers. It also promoted the National Law on Food and Nutrition Security, which obliged the federal government to uphold people’s right to food and create food councils at more localised levels. Along with cash transfers given to poor mothers and an increase in the minimum wage, these reforms lifted millions of people out of chronic undernourishment. The Zero Hunger programme could claim real success. In contrast to Algeria, diverse groups in Brazilian society – including teachers, farmers, clergy and health professionals – were able to play a more proactive role in national food politics. Indeed, their collective contribution also reshaped international policy. When the minister for food security in the Lula government, José Graziano da Silva, was elected to the head of the UN’s Food and Agricultural Organisation in 2011, he began to promote many of the same policies that had been developed in Brazil. A twin-track strategy based on investments in rural areas to boost the incomes of farming families and basic welfare payments to protect the most vulnerable in society was advocated.
Backed by other United Nations agencies and the UN Secretary-General, Ban Ki-moon, over the next three years Zero Hunger Challenge programmes were launched in a number of countries including St. Lucia, Laos and Zambia. This approach also informed the 2015 UN Sustainable Development Goals, which set out a roadmap for the end of world hunger by 2030. That said, it is a lot easier to make policies and plans than to achieve them. Key in the Brazilian case was the mobilisation of national civil society, which brought forward people willing to play a role in political affairs. In countries where this is not encouraged, it is hard to see plans for the reduction of poverty and hunger taking effect. Moreover, Brazil itself is far from perfect, with mass protests and political upheavals in 2016 reflecting the nation’s slide into ever-deepening recession. Chronic hunger may have diminished but temporary hunger and poor diet remain, especially in the impoverished areas of Northeast Brazil and among indigenous communities. Ensuring their right to food is an ongoing struggle, and one that will have to overcome the significant domestic political and economic challenges that Brazil faces.


5.4A: Protection
The skin provides an overlaying protective barrier from the environment and pathogens while contributing to the adaptive immune system.
LEARNING OBJECTIVE
Describe the ways in which the integumentary system protects the body
Key Takeaways
Key Points
- The skin provides a protective barrier from the external environment and prevents dehydration.
- Langerhans cells in the skin also contribute to protection as they are part of the adaptive immune system.
- The integumentary system protects the body’s internal living tissues and organs, protects against invasion by infectious organism, and protects the body from dehydration.
Key Terms
- vitamin D : An important vitamin synthesized thanks to the skin.
- melanocytes : Cells that help protect our body from radiological damage.
- Langerhans cells : Langerhans cells are dendritic cells (antigen-presenting immune cells) of the skin and mucosa that contain large granules.
The skin helps protect our body’s internal structures from physical, chemical, biological, radiological, and thermal damage as well as damage from starvation and malnutrition.
Physical and Chemical Damage
Human skin: A diagram of human skin.
The skin is composed of tough skin cells as well as a tough protein called keratin that guard tissues, organs, and structures underneath the skin against physical damage from minor cuts, scratches, and abrasions. Because our skin is tough and largely waterproof, it helps protect internal structures from chemical irritants such as man-made detergents or even natural irritants like poison ivy.
Otherwise, these dangerous chemicals would seep into our sensitive internal environment. The waterproof nature of our skin also ensures that important molecules stay within our body.
Biological Damage
The skin also contains important cells called Langerhans cells. These cells help our immune system fight off infectious biological agents, like bacteria that try to get further into our body through skin that may have been compromised by physical damage.
Sebaceous glands associated with the skin secrete substances that help fight off potentially dangerous microorganisms as well. These glands also secrete substances that help keep our skin hydrated, and thus more resistant to bacterial invasion.
Radiological Damage
Our skin also contains melanocytes that produce a pigment called melanin. This protects the body from radiological damage via the sun’s UV radiation (or that from tanning beds).
Other Protective Roles
Part of our skin is made up of fat. This fat serves three large purposes:
- It helps cushion internal structures against any physical blows.
- It acts as a food source, protecting our body from the effects of starvation.
- It helps insulate us against cold temperatures.
Our skin is also closely associated with sweat glands that help protect us from high temperatures by cooling us off through the process of evaporation. These glands also help to excrete potentially dangerous substances, like urea, out of the body.
All sorts of sensory receptors are found within the skin as well. These help move our body parts away from potential sources of damage, like hot stoves, when they sense danger, thereby protecting our body from great harm.
Finally, the skin is also important for the synthesis of vitamin D, which is an important vitamin for the building of strong and healthy bones. Ergo, the skin protects the body from fractures if we do not otherwise get enough of this vitamin from food-based sources.


1.27: Cosmic Microwave Background Anisotropies
Introduction
The plasma that existed from the first fractions of a second of the Big Bang until it transitioned to a neutral gas 380,000 years later is a beautiful gift of nature. Gently disturbed away from equilibrium by mysterious, very early-universe processes, the plasma is an unusually simple, natural dynamical system. Due to its simplicity, we can calculate the evolution of these disturbances with high accuracy and use these calculations to predict observable consequences. The agreement between our predictions and precision observations is extraordinary, allowing us a high degree of confidence that we understand in detail events that were transpiring 14 billion years ago, throughout the cosmos, as far away as the edge of the observable universe.
The simplicity of this system is perhaps best understood in contrast to other natural systems. Naturally occurring phenomena are usually too complicated to understand fully from first principles. We know the underlying physics, but the equations are really complicated to solve, and we’d need an absurd amount of data to narrow down from the set of all possible solutions of the equations, to the one that is actually being realized with the phenomenon in question. Examples of complicated phenomena are supernova explosions, earthquakes, and human behavior. We think we know the underlying physics at play, but getting from there to predictions for observations is extremely difficult, if not impossible.
Usually, to observe a system for which we can make predictions, we need to carefully construct the system ourselves. We call such carefully constructed scenarios, ‘experiments.’ Although the exception rather than the rule, there are examples of naturally occurring systems that are amenable to our understanding. The solar system is such a naturally occurring system, with its simplicity arising from the fact that gravity is the dominantly important force, and the high accuracy of the simplifying approximation of the planets and stars as point masses. The simplicity of this system, and the regularity of the passage of the planets and Sun across the sky may have, in fact, played an important role in the discovery of science itself, as argued by Steven Weinberg in To Explain the World: The Discovery of Modern Scienc e.
Another such simple system is this early plasma, often called the primordial plasma. We are motivated to share with you our observations of the sky at millimeter to sub-millimeter wavelengths (observations that reveal patterns in this plasma just as it disappeared about 14 billion years ago) and the remarkable agreement of the statistical properties of these observations with our calculations. To fully appreciate this agreement, we will introduce you to some fundamental aspects of how those calculations are done.
In this chapter, we will introduce the primordial plasma and how observing the sky in millimeter to sub-millimeter wavelengths allows us to see the plasma just as it disappears by transitioning to a neutral gas. In this section we will also see the extraordinary agreement between the statistical properties of maps of the CMB and our predictions for these statistical properties, predictions that follow from our understanding of the dynamics of the plasma. The main focus of this chapter is to gain a qualitative understanding of a statistical property of CMB maps called a 'power spectrum.' In the next chapter, we will model the plasma as a fluid and introduce a simple version of the dynamical equation that governs the evolution of its density, a wave equation. We will find solutions to the wave equation, and demonstrate the benefits of Fourier decomposition for evolving a system governed by such an equation. Then we will work out the prediction of the main qualitative feature of the spectrum -- the existence of a series of peaks.
Before we fully begin, let me digress with one bit of history here related to the prediction of this series of peaks and their detection. On the left is a photo I took at the June 30, 2001 launch of the Wilkinson Microwave Anisotropy Probe (WMAP). From left to right are four fellow theorists: Ned Wright, Neil Cornish, Dick Bond, and Rashid Sunyaev. Despite Rashid’s significant contributions to the field many of us, including me, had not met him before. Dick Bond was ever-present at Rashid’s side, introducing him to people saying, "This is Rashid Sunyaev, as in Sunyaev-Zel'dovich" like he was bringing you in on some funny joke, as everyone in my field knows of something we call the Sunyaev-Zel’dovich effect. One couldn't help but smile in response. Rashid, meanwhile, was quite emotional, telling me that his advisor (Zel'dovich) had told him, "This is beautiful physics, but it will never be observed.  I was so ignorant of the history, I only realized later he was talking about their 1967 work that was the first to point out that there would be acoustic peaks in the power spectrum of the CMB. We've now measured those peaks with very high precision. The WMAP satellite launched that day worked spectacularly well and was a big step forward in our measurements of the CMB. The launch was a joyful occasion.
The Primordial Plasma
A plasma is a state of matter in which electrons are dissociated from nuclei. For example, the Sun is a great ball of gravitationally-bound plasma. Plasmas are hot, as it is the thermal energy that keeps the negatively charged electrons from binding with the positively charged nuclei. Plasmas inevitably include photons, since the charged particles (protons and electrons) accelerate in the electric fields of other charged particles and radiate photons as a result. The primordial plasma was a collection of electrons, hydrogen and helium nuclei, trace amounts of other light nuclei, and photons. It existed before there were any stars, and so before there were any heavier elements that eventually were formed in stars. The formation of helium and other light elements is the subject of the classic popular book by Steven Weinberg, The First Few Minutes , to which we allude with the title of this chapter.
As we have discussed before, the plasma eventually cools sufficiently that the electrons bind with the protons and helium nuclei, so that the plasma transforms into a neutral gas. The universe becomes transparent and the thermal photons start freely streaming across the universe. Those photons that originated at just the right distance from us, and were headed our way, are arriving now. They give us an image of what the universe was like at the time of this transition in a thin spherical shell around us at a distance of about 46 billion light years that we call the last-scattering surface. Thus we have a means of studying the primordial plasma observationally. We saw maps of the CMB radiation in the previous chapter that revealed the monopole and the dipole. If we remove the monopole and the dipole then the remaining map is dominated by features that correspond to inhomogeneities on this last-scattering surface. A full-sky map of the CMB, as determined by data from the Planck satellite launched in 2009, is shown below. Planck flew later than WMAP but had higher angular resolution, increased sensitivity, and mapped the sky over a broader range of frequencies.
The following image is a projection of the spherical sky onto a 2D map. To better understand how this relates to what we observe from Earth, explore the virtual CMB planetarium below it by clicking and dragging. This shows how the CMB would look over a tree-lined horizon, if human eyes were extremely sensitive to light at millimeter wavelengths.
'
CMB Planetarium
The CMB Power Spectrum
So how do we learn things from a map of the CMB? Our theories do not actually predict the map. They predict statistical properties of the map. The most important statistical property is called the "power spectrum." The power spectrum tells us about the smoothness/roughness of the map, as a function of angular scale.
Qualitative Description
Now what does that mean, "as a function of angular scale?" Well, how would you describe the surface of the Pacific Ocean? Is it smooth or rough? The answer depends on length scale. On very large scales, scales much larger than the typical wavelength of a wave, it is quite smooth. But then on scales the length of a typical wave it is rougher. Zooming in further, to length scales smaller than a typical wave wavelength, it might appear smooth again.
Here, we'd also like to make very clear that we are not talking about the wavelengths of radiation emitted from the CMB. The power spectrum of the CMB refers to spatial wavelength of fluctuations in the temperature of the CMB across the sky.
We can do the same type of analysis with a map of the cosmic microwave background that we did with the Pacific Ocean. The scale-dependent measure of "roughness" we call "power." The power spectrum of the CMB is shown in the left panel below. The power is on the y-axis and the angular scale is shown on the x-axis. A higher multipole moment \( \ell \) corresponds to a smaller scale. On the right panel is a simulated map that is consistent with this power spectrum. (Note that in all the following animations and pictures, each square shows an 8.5 x 8.5 degree patch of the CMB).
The y axis is power and the x axis is a measure of angular scale with larger scales on the left and smaller scales on the right.
Quantitative Description
Now let's give a more mathematical description of the power spectrum. Any function on the surface of a sphere, \(T(\theta, \phi)\) can be written as a sum over complex functions on the sphere with well-defined wavelengths, the spherical harmonics \(Y_{lm}(\theta,\phi)\):
\[T(\theta,\phi) = \sum_{lm} a_{lm} Y_{lm}(\theta,\phi).\]
The sum over \(l\) ranges from 0 to infinity and the sum over \(m\) ranges from \(-l\) to \(+l\). The \(l\) value tells you the wavelength of the oscillations: \(Y_{lm}(\theta,\phi)\) undergoes \(l\) oscillations as one runs through all 360 degrees of a great circle. The spherical harmonics for the lowest values of \(l\) have names: monopole, dipole, quadrupole and octupole are for \(l=0,1,2,3\) respectively. The figure below shows examples of dipole, quadrupole and octupole patterns over the whole sky (using the Mollweide projection for mapping the sphere on to a plane). Notice how larger multipoles like \( \ell = 3 \) correspond to smaller features on the CMB map, whereas lower multipoles like \( \ell = 1\) correspond to very large scale features.
Our cosmological models do not predict the temperature in a particular direction, or the value of a particular \(a_{lm}\); they predict statistical properties such as the mean and variance of \(a_{lm}\). For isotropic cosmologies, the mean of all except for the monopole is zero. The variance in isotropic theories is independent of \(m\) and given by
\[C_l \equiv \langle a_{lm}a^*_{lm} \rangle.\]
This variance, as a function of \(l\), is called the power spectrum. Rather than \(C_l\), we usually plot \(D_l \equiv l(l+1)C_l/(2\pi)\) instead.
More Intuition about the Power Spectrum
Below, we show the same power spectrum and CMB map from above, but only features of the map whose scale falls within the grey box are included. At first, we see large scale features that increase in amplitude as the grey box passes over the first peak in the power spectrum at \(\ell\) of about 200. We then see progressively smaller features of lower amplitude as the box scans through the higher-\(\ell\) region. The pitch of the audio corresponds to the size of the features and the volume to the amplitude of the power spectrum in that range. We can hear the pitch increasing as we move from left to right. The volume rises as we pass over the first peak, then tapers off.
In the video below, the grey box steadily includes more of the power spectrum over time. As the box expands, we can hear more frequencies and see more features on the map.
Play "Match the Power Spectrum to the map"
First, observe each simulated CMB map. Next, click and drag to match the letter of the CMB map to the correct power spectrum on the left. Remember, large features correspond to power on the left of the power spectrum and small features correspond to power on the right side of the power spectrum. Feel free to refer back to the above videos. Warning: this is challenging. You will have to put some thought into it, and even then are likely to get it wrong the first time. If you do, think again and try again. I incorrectly assigned two of the power spectra the first time I tried it!
Matching Game
Comparison with Observations
Observations of the microwave background intensity as a function of direction on the sky have been carried out since the late sixties from telescopes on the ground, on high-altitude balloons, and from three different spacecraft (COBE, WMAP, and Planck). It is useful to get above the atmosphere as the atmosphere itself emits and absorbs at millimeter to submillimeter wavelengths, so balloons and spacecraft offer advantages. Ground-based instruments have been competitive though, in particular at higher angular resolution (higher \(\ell\)), which requires larger telescopes that are harder to lift above the atmosphere. One of the best ground-based sites for observing at millimeter to submillimeter wavelengths is at the South Pole, where the 10-meter South Pole Telescope (SPT) has operated since 2007.
The WMAP satellite, like Planck, observed the full sky, whereas the SPT survey covered 2,500 square degrees, about 6% of the sky, but with lower noise and higher angular resolution. The figure below is from Story et al. (2013), showing the determination of the CMB temperature power spectrum from 7 years of WMAP observations and from the complete SPT 2500 sq. degree survey. The solid line is a fit to these spectra that includes the contribution from the CMB itself assuming the best-fit standard cosmological model (dashed line), and the contribution from other extragalactic sources of radiation.
The figure below is from the Planck Collaboration showing their determination of the CMB power spectrum and the predicted power spectrum of the standard cosmological model with its parameters adjusted to best agree with the data. The lower panel shows the 'residuals': the data points after the best-fit model has been subtracted from it. It allows one to better inspect the quality of the agreement between theory and data. The small amplitude (relative to the error bars) of the departures from zero in the lower panel indicate agreement between theory and observation. Note that the y axis for the residuals is different for \(\ell < 30\) than it is for \(\ell > 30\).
The agreement between observation and theory here is quite extraordinary. It gives us a very high level of confidence that we have a good understanding of the physical conditions of the universe when the scale factor was over 1,000 times smaller than it is today, around 14 billion years ago.
The Progression of CMB Power Spectrum Experiments
The following video tracks the progress of CMB measurements from 1967 to 2020. Gray boxes show the weighted average of prior measurements. The error bars indicate the type and quality of a measurement. The horizontal bar indicates the range of multipoles \( \ell \) over which the power measurement was taken. The vertical bar indicates the \( 1 \sigma \) error. (That means there is a 68% probability the true power lies within the vertical bar). If the error bar has a downward pointing arrow, it indicates that the experiment only determined an upper bound on the CMB power, not a detection.
Video Quiz