22.4: Exercises
Calculate each of the following.
- \(\displaystyle [\mathrm{GF}(3^6) : \mathrm{GF}(3^3)]\)
- \(\displaystyle [\mathrm{GF}(128): \mathrm{GF}(16)]\)
- \(\displaystyle [\mathrm{GF}(625) : \mathrm{GF}(25) ]\)
- \(\displaystyle [\mathrm{GF}(p^{12}): \mathrm{GF}(p^2)]\)
Calculate \([\mathrm{GF}(p^m): \mathrm{GF}(p^n)]\text{,}\) where \(n \mid m\text{.}\)
What is the lattice of subfields for \(\mathrm{GF}(p^{30})\text{?}\)
Let \(\alpha\) be a zero of \(x^3 + x^2 + 1\) over \({\mathbb Z}_2\text{.}\) Construct a finite field of order \(8\text{.}\) Show that \(x^3 + x^2 + 1\) splits in \({\mathbb Z}_2(\alpha)\text{.}\)
Construct a finite field of order \(27\text{.}\)
Prove or disprove: \({\mathbb Q}^\ast\) is cyclic.
Factor each of the following polynomials in \({\mathbb Z}_2[x]\text{.}\)
- \(\displaystyle x^5- 1\)
- \(\displaystyle x^6 + x^5 + x^4 + x^3 + x^2 + x + 1\)
- \(\displaystyle x^9 - 1\)
- \(\displaystyle x^4 +x^3 + x^2 + x + 1\)
Prove or disprove: \({\mathbb Z}_2[x] / \langle x^3 + x + 1 \rangle \cong {\mathbb Z}_2[x] / \langle x^3 + x^2 + 1 \rangle\text{.}\)
Determine the number of cyclic codes of length \(n\) for \(n = 6, 7, 8, 10\text{.}\)
Prove that the ideal \(\langle t + 1 \rangle\) in \(R_n\) is the code in \({\mathbb Z}_2^n\) consisting of all words of even parity.
Construct all BCH codes of
- length \(7\text{.}\)
- length \(15\text{.}\)
Prove or disprove: There exists a finite field that is algebraically closed.
Let \(p\) be prime. Prove that the field of rational functions \({\mathbb Z}_p(x)\) is an infinite field of characteristic \(p\text{.}\)
Let \(D\) be an integral domain of characteristic \(p\text{.}\) Prove that \((a - b)^{p^n} = a^{p^n} - b^{p^n}\) for all \(a, b \in D\text{.}\)
Show that every element in a finite field can be written as the sum of two squares.
Let \(E\) and \(F\) be subfields of a finite field \(K\text{.}\) If \(E\) is isomorphic to \(F\text{,}\) show that \(E = F\text{.}\)
Let \(F \subset E \subset K\) be fields. If \(K\) is a separable extension of \(F\text{,}\) show that \(K\) is also separable extension of \(E\text{.}\)
Let \(E\) be an extension of a finite field \(F\text{,}\) where \(F\) has \(q\) elements. Let \(\alpha \in E\) be algebraic over \(F\) of degree \(n\text{.}\) Prove that \(F( \alpha )\) has \(q^n\) elements.
Show that every finite extension of a finite field \(F\) is simple; that is, if \(E\) is a finite extension of a finite field \(F\text{,}\) prove that there exists an \(\alpha \in E\) such that \(E = F( \alpha )\text{.}\)
Show that for every \(n\) there exists an irreducible polynomial of degree \(n\) in \({\mathbb Z}_p[x]\text{.}\)
Prove that the Frobenius map \(\Phi : \mathrm{GF}(p^n) \rightarrow \mathrm{GF}(p^n)\) given by \(\Phi : \alpha \mapsto \alpha^p\) is an automorphism of order \(n\text{.}\)
Show that every element in \(\mathrm{GF}(p^n)\) can be written in the form \(a^p\) for some unique \(a \in \mathrm{GF}(p^n)\text{.}\)
Let \(E\) and \(F\) be subfields of \(\mathrm{GF}(p^n)\text{.}\) If \(|E| = p^r\) and \(|F| = p^s\text{,}\) what is the order of \(E \cap F\text{?}\)
Wilson's Theorem
Let \(p\) be prime. Prove that \((p-1)! \equiv -1 \pmod{p}\text{.}\)
If \(g(t)\) is the minimal generator polynomial for a cyclic code \(C\) in \(R_n\text{,}\) prove that the constant term of \(g(x)\) is \(1\text{.}\)
Often it is conceivable that a burst of errors might occur during transmission, as in the case of a power surge. Such a momentary burst of interference might alter several consecutive bits in a codeword. Cyclic codes permit the detection of such error bursts. Let \(C\) be an \((n,k)\)-cyclic code. Prove that any error burst up to \(n-k\) digits can be detected.
Prove that the rings \(R_n\) and \({\mathbb Z}_2^n\) are isomorphic as vector spaces.
Let \(C\) be a code in \(R_n\) that is generated by \(g(t)\text{.}\) If \(\langle f(t) \rangle\) is another code in \(R_n\text{,}\) show that \(\langle g(t) \rangle \subset \langle f(t) \rangle\) if and only if \(f(x)\) divides \(g(x)\) in \({\mathbb Z}_2[x]\text{.}\)
Let \(C = \langle g(t) \rangle\) be a cyclic code in \(R_n\) and suppose that \(x^n - 1 = g(x) h(x)\text{,}\) where \(g(x) = g_0 + g_1 x + \cdots + g_{n - k} x^{n - k}\) and \(h(x) = h_0 + h_1 x + \cdots + h_k x^k\text{.}\) Define \(G\) to be the \(n \times k\) matrix
\[ G = \begin{pmatrix} g_0 & 0 & \cdots & 0 \\ g_1 & g_0 & \cdots & 0 \\ \vdots & \vdots &\ddots & \vdots \\ g_{n-k} & g_{n-k-1} & \cdots & g_0 \\ 0 & g_{n-k} & \cdots & g_{1} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & g_{n-k} \end{pmatrix} \nonumber \]
and \(H\) to be the \((n-k) \times n\) matrix
\[ H = \begin{pmatrix} 0 & \cdots & 0 & 0 & h_k & \cdots & h_0 \\ 0 & \cdots & 0 & h_k & \cdots & h_0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ h_k & \cdots & h_0 & 0 & 0 & \cdots & 0 \end{pmatrix}\text{.} \nonumber \]
- Prove that \(G\) is a generator matrix for \(C\text{.}\)
- Prove that \(H\) is a parity-check matrix for \(C\text{.}\)
- Show that \(HG = 0\text{.}\)


4.2.1: Oxidation and Reduction
Oxidation-Reduction Reactions
When a transfer of electrons occurs between a metal and a nonmetal, we can view this as an example of oxidation-reduction reaction (or redox reaction).
Redox reactions are a broad class of chemical reactions, but some redox reactions involve the transfer of electrons between reactant species to yield ionic compounds, such as the reaction between sodium and chlorine to yield sodium chloride:
\[\ce{2Na}(s)+\ce{Cl2}(g)\rightarrow \ce{2NaCl}(s) \nonumber \]
It is helpful to view the process with regard to each individual reactant, that is, to represent the fate of each reactant in the form of an equation called a half-reaction :
\[ \begin{align*} \ce{2Na}(s) &\rightarrow \ce{2Na+}(s)+\ce{2e-} \\[4pt] \ce{Cl2}(g)+\ce{2e-} &\rightarrow \ce{2Cl-}(s) \end{align*} \nonumber \]
These equations show that Na atoms lose electrons while Cl atoms (in the Cl 2 molecule) gain electrons , the “ s ” subscripts for the resulting ions signifying they are present in the form of a solid ionic compound. For redox reactions of this sort, the loss and gain of electrons define the complementary processes that occur:
\(\begin{align}
\textbf{oxidation}&=\textrm{loss of electrons}\\
\textbf{reduction}&=\textrm{gain of electrons}
\end{align}\)
In this reaction, then, sodium is oxidized and chlorine undergoes reduction . Viewed from a more active perspective, sodium functions as a reducing agent (reductant) , since it provides electrons to (or reduces) chlorine. Likewise, chlorine functions as an oxidizing agent (oxidant) , as it effectively removes electrons from (oxidizes) sodium.
\(\begin{align}
\textbf{reducing agent}&=\textrm{species that is oxidized}\\
\textbf{oxidizing agent}&=\textrm{species that is reduced}
\end{align}\)


  This is often challenged by those who argue that certain identities are simply a temporary “phase” or that queerness is unnatural. These arguments and views will be addressed in this chapter, as well as a section on terms that have been coined by scholars in queer theory and other academic fields that help us understand sexuality more.
Before we continue, an important thing to understand is that this information will likely change, and a certain definition may not be what everyone agrees on. People might share a label, but their experiences can vary in several ways, and some people choose to not label themselves at all when it comes to their attractions. The way these people present themselves can be drastically different, and their preferences in their partners can vary as well. The essence of this is to say an identity is not a “one size fits all” box, and rather than looking for external validation around a label, we hope this chapter helps you decide that you get to define yourself in whatever way you like.
When you think of a potential significant other (or if you already have one), what quality is most important to you? Your answer may differ from others or it may be similar. As we will learn later on in this book, attraction does fall down similar patterns but ultimately we all like who we like based on multiple factors that are unique to who we are. While you may have some preferences when it comes to physical features, people generally want their partner to have similar values, and certain qualities are perceived to be more important than others.
Specific identities will have certain stereotypes and misconceptions associated with them. These stereotypes come from cultural labels surrounding differing sexual orientations often perpetuated in the media. In a 2010 study entitled Fairy tales: Attraction and stereotypes in same-gender relationships , Felmlee et al discuss common stereotypes surrounding same-gender relationships such as “exhibition of gender atypical traits, sexual promiscuity, and predatory sexual tendencies,” and challenge these ideas with their data, which suggests that the reality is not as simple as previously believed, and that these stereotypes serve to create false beliefs surrounding same sex relationships. Just as heterosexual people have their own preferences when looking for a significant other, queer people have the same and the preferences are varied and multifaceted (Felmlee et al, 2010).


9.9D: Muscles that Cause Movement at the Ankle
Muscles of the leg insert into ankle and foot bones to facilitate ankle movement.
- Describe the muscles that cause the ankle to move
Key Points
- The ankle consists of two joints which permit dorsiflexion, plantarflexion, inversion, and eversion of the foot.
- Strong ligaments hold the ankle joint in place, although it is susceptible to damage.
- Muscles controlling movement at the ankle are found in the leg and can be split into anterior, posterior, and lateral compartments.
Key Terms
- plantarflexion : Movement of the foot downwards away from the lower leg.
- eversion : Tilting of the foot so the sole faces away from the midline.
- inversion : Tilting of the foot so the sole faces into the midline.
- dorsiflexion : Movement of the foot upwards towards the lower leg.
Movement at the ankle is controlled by two joints. The ankle or talocrural joint is formed from the tibia and fibula of the lower leg and talus of the foot. Functionally, it acts as a hinge, allowing dorsiflexion (pulling the foot upwards towards the lower leg) and plantarflexion (pulling the foot downwards away from the lower leg). Eversion (tilting of the sole of the foot away from the midline) and inversion (tilting of the sole of the foot inwards towards the midline) is controlled by the subtalar joint formed between the talus and calcaneus bones of the foot.
The ankle joint is held in place by numerous strong ligaments that can be easily damaged when excessive force is placed on the ankle, particularly during strenuous inversion and eversion. Movement at the ankle is key for maintenance of posture and balance, but is most important in locomotion. Variation in muscle activation can control the movement of the ankle joint, allowing the foot to generate graduated force.
Muscles that generate movement at the ankle are generally found in the lower leg and can be split into three categories.
Anterior Compartment
(a) Anterior Compartment of the Leg and (b) Posterior Compartment of the leg: Anterior view of leg showing the muscles and tendons involved in ankle movement. : Posterior view of leg showing muscles and tendons involved in ankle movement.
Three muscles in the anterior compartment of the leg act to dorsiflex and invert the foot at the ankle joint.
-
Tibialis Anterior
: The tibialis anterior muscle is located alongside the lateral surface of the tibia and is the strongest dorsiflexor of the foot.
- Attachments: Originates from the lateral surface of the tibia and attaches to the base of the big toe.
- Actions: Dorsiflexion and inversion of the foot.
-
Extensor Digitorum Longus
: The extensor digitorum longus is a deep-lying extrinsic muscle that runs the length of the tibia.
- Attachments: Originates from the tibia and transitions into a tendon, passes into the foot, splits into four, and attaches to the toes.
- Actions: Extension of the toes and dorsiflexion of the foot.
-
Extensor Hallucis Longus
: The extensor hallucis longus is a deep lying extrinsic muscle beneath the extensor digitorum longus.
- Attachments: Originates from the fibula and attaches to the big toe.
- Actions: Extension of the big toe, and dorsiflexion of the foot.
Posterior Compartment
Several muscles are located in the posterior compartment of the leg, typically grouped into superficial and basal groups. The majority of these muscles work to plantarflex the foot at the ankle.
Superficial Muscles
The superficial muscles give rise to the characteristic shape of the lower leg.
-
Gastrocnemius
: The gastrocnemius, a two-headed muscle, is the most superficial of the muscles in the posterior compartment.
- Attachments: Both heads originate from the femur. The fibers converge to form the calcaneal tendon which attaches to the heel.
- Actions: Plantarflexes the foot, can also flex the lower leg at the knee but is not key in this movement.
-
Plantaris
: The plantaris is a small muscle lying between the gastrocnemius and soleus. It is absent in 10% of people.
- Attachments: Originates from the femur and attaches to the heel via the calcaneal tendon.
- Actions: Plantarflexes the foot, can also flex the lower leg at the knee but is not key in this movement.
-
Soleus
: The soleus is a large flat muscle which is the deepest lying of the superficial muscles.
- Attachments: Originates from the tibia and fibula and attaches to the heel via the calcaneal tendon.
- Actions: Plantarflexes the foot.
Deep Muscles
-
Tibialis Posterior
: The tibialis posterior is the deepest lying of the muscles in the posterior compartment.
- Attachments: Originates from the tibia and fibula and attaches to the plantar surfaces of the toes.
- Actions: Inverts and plantarflexes the foot, maintains the arch of the foot.
Lateral Compartment
Two muscles found in the lateral compartment function to control eversion of the foot. Physiologically, there is a preference for the foot to invert, so these muscles also prevent excessive inversion.
-
Fibularis Longus
: The fibularis longus is the longer and more superficial of the two muscles.
- Attachments: Originates from the fibula and tibia. The fibers converge into a tendon which passes under the foot and attaches to the medial side of the foot.
- Actions: Eversion and plantarflexion of the foot.
-
Fibularis Brevis
: The fibularis brevis muscles is the deeper and shorter of the two muscles.
- Attachments: Originates from the lateral surface of the fibula and attaches to the little toe.
- Actions: Eversion of the foot.
KEY MOVEMENTS
- Eversion of the Foot (tilting of the sole of the foot away from the midline): Performed by the fibularis brevis and fibularis longus.
- Inversion of the Foot (tilting of the sole of the foot inwards towards the midline ) : Performed by the tibialis posterior and tibialis anterior.
- Dorsiflexion of the Foot (pulling the foot upwards towards the leg): Performed by the tibialis anterior, extensor hallucis longus and extensor digitorum longus.
- Plantarflexion of the Foot (pulling the foot downwards away from the lower leg): Performed by the gastrocnemius, plantaris, soleus and fibularis longus.


5.5E: Green's Theorem (Exercises)
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For the following exercises, evaluate the line integrals by applying Green’s theorem.
1. \(\displaystyle \int_C 2xy\,dx+(x+y)\,dy\), where \(C\) is the path from \((0, 0)\) to \((1, 1)\) along the graph of \(y=x^3\) and from \((1, 1)\) to \((0, 0)\) along the graph of \(y=x\) oriented in the counterclockwise direction
2. \(\displaystyle \int_C 2xy\,dx+(x+y)\,dy\), where \(C\) is the boundary of the region lying between the graphs of \(y=0\) and \(y=4−x^2\) oriented in the counterclockwise direction
- Answer
- \(\displaystyle \int_C2xy\,dx+(x+y)\,dy=\frac{32}{3}\) units of work
3. \(\displaystyle \int_C 2\arctan\left(\frac{y}{x}\right)\,dx+\ln(x^2+y^2)\,dy\), where \(C\) is defined by \(x=4+2\cos θ,\;y=4\sin θ\) oriented in the counterclockwise direction
4. \(\displaystyle \int_C \sin x\cos y\,dx+(xy+\cos x\sin y)\,dy\), where \(C\) is the boundary of the region lying between the graphs of \(y=x\) and \(y=\sqrt{x}\) oriented in the counterclockwise direction
- Answer
- \(\displaystyle \int_C\sin x\cos y\,dx+(xy+\cos x\sin y)\,dy=\frac{1}{12}\) units of work
5. \(\displaystyle \int_C xy\,dx+(x+y)\,dy\), where \(C\) is the boundary of the region lying between the graphs of \(x^2+y^2=1\) and \(x^2+y^2=9\) oriented in the counterclockwise direction
6. \(\displaystyle ∮_C (−y\,dx+x\,dy)\), where \(C\) consists of line segment \(C_1\) from \((−1,0)\) to \((1, 0)\), followed by the semicircular arc \(C_2\) from \((1, 0)\) back to \((-1, 0)\)
- Answer
- \(\displaystyle ∮_C (−y\,\,dx+x\,\,dy)=π\) units of work
For the following exercises, use Green’s theorem.
7. Let \(C\) be the curve consisting of line segments from \((0, 0)\) to \((1, 1)\) to \((0, 1)\) and back to \((0, 0)\). Find the value of \(\displaystyle \int_C xy\,dx+\sqrt{y^2+1}\,dy\).
8. Evaluate line integral \(\displaystyle \int_C xe^{−2x}\,dx+(x^4+2x^2y^2)\,dy\), where \(C\) is the boundary of the region between circles \(x^2+y^2=1\) and \(x^2+y^2=4\), and is a positively oriented curve.
- Answer
- \(\displaystyle \int_C xe^{−2x}\,dx+(x^4+2x^2y^2)\,dy=0\) units of work
9. Find the counterclockwise circulation of field \(\vecs F(x,y)=xy\,\mathbf{\hat i}+y^2\,\mathbf{\hat j}\) around and over the boundary of the region enclosed by curves \(y=x^2\) and \(y=x\) in the first quadrant and oriented in the counterclockwise direction.
10. Evaluate \(\displaystyle ∮_C y^3\,dx−x^3y^2\,dy\), where \(C\) is the positively oriented circle of radius \(2\) centered at the origin.
- Answer
- \(\displaystyle ∮_C y^3\,dx−x^3y^2\,dy=−20π\) units of work
11. Evaluate \(\displaystyle ∮_C y^3\,dx−x^3\,dy\), where \(C\) includes the two circles of radius \(2\) and radius \(1\) centered at the origin, both with positive orientation.
12. Calculate \(\displaystyle ∮_C −x^2y\,dx+xy^2\,dy\), where \(C\) is a circle of radius \(2\) centered at the origin and oriented in the counterclockwise direction.
- Answer
- \(\displaystyle ∮_C −x^2y\,dx+xy^2\,dy=8π\) units of work
13. Calculate integral \(\displaystyle ∮_C 2[y+x\sin(y)]\,dx+[x^2\cos(y)−3y^2]\,dy\) along triangle \(C\) with vertices \((0, 0), \,(1, 0)\) and \((1, 1)\), oriented counterclockwise, using Green’s theorem.
14. Evaluate integral \(\displaystyle ∮_C (x^2+y^2)\,dx+2xy\,dy\), where \(C\) is the curve that follows parabola \(y=x^2\) from \((0,0), \,(2,4),\) then the line from \((2, 4)\) to \((2, 0)\), and finally the line from \((2, 0)\) to \((0, 0)\).
- Answer
- \(\displaystyle ∮_C (x^2+y^2)\,dx+2xy\,dy=0\) units of work
15. Evaluate line integral \(\displaystyle ∮_C (y−\sin(y)\cos(y))\,dx+2x\sin^2(y)\,dy\), where \(C\) is oriented in a counterclockwise path around the region bounded by \(x=−1, \,x=2, \,y=4−x^2\), and \(y=x−2.\)
For the following exercises, use Green’s theorem to find the area.
16. Find the area between ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) and circle \(x^2+y^2=25\).
- Answer
- \(A=19π\;\text{units}^2\)
17. Find the area of the region enclosed by parametric equation
\(\vecs p(θ)=(\cos(θ)−\cos^2(θ))\,\mathbf{\hat i}+(\sin(θ)−\cos(θ)\sin(θ))\,\mathbf{\hat j}\) for \(0≤θ≤2π.\)
18. Find the area of the region bounded by hypocycloid \(\vecs r(t)=\cos^3(t)\,\mathbf{\hat i}+\sin^3(t)\,\mathbf{\hat j}\). The curve is parameterized by \(t∈[0,2π].\)
- Answer
- \(A=\frac{3}{8π}\;\text{units}^2\)
19. Find the area of a pentagon with vertices \((0,4), \,(4,1), \,(3,0), \,(−1,−1),\) and \((−2,2)\).
20. Use Green’s theorem to evaluate \(\displaystyle \int_{C^+}(y^2+x^3)\,dx+x^4\,dy\), where \(C^+\) is the perimeter of square \([0,1]×[0,1]\) oriented counterclockwise.
- Answer
- \(\displaystyle \int_{C^+} (y^2+x^3)\,dx+x^4\,dy=0\)
21. Use Green’s theorem to prove the area of a disk with radius \(a\) is \(A=πa^2\;\text{units}^2\).
22. Use Green’s theorem to find the area of one loop of a four-leaf rose \(r=3\sin 2θ\). ( Hint : \(x\,dy−y\,dx=r^2\,dθ\)).
- Answer
- \(A=\frac{9π}{8}\;\text{units}^2\)
23. Use Green’s theorem to find the area under one arch of the cycloid given by the parametric equations: \(x=t−\sin t,\;y=1−\cos t,\;t≥0.\)
24. Use Green’s theorem to find the area of the region enclosed by curve
\(\vecs r(t)=t^2\,\mathbf{\hat i}+\left(\frac{t^3}{3}−t\right)\,\mathbf{\hat j},\) for \(−\sqrt{3}≤t≤\sqrt{3}\).
- Answer
- \(A=\frac{8\sqrt{3}}{5}\;\text{units}^2\)
25. [T] Evaluate Green’s theorem using a computer algebra system to evaluate the integral \(\displaystyle \int_C xe^y\,dx+e^x\,dy\), where \(C\) is the circle given by \(x^2+y^2=4\) and is oriented in the counterclockwise direction.
26. Evaluate \(\displaystyle \int_C(x^2y−2xy+y^2)\,ds\), where \(C\) is the boundary of the unit square \(0≤x≤1,\;0≤y≤1\), traversed counterclockwise.
- Answer
- \(\displaystyle \int_C (x^2y−2xy+y^2)\,ds=3\)
27. Evaluate \(\displaystyle \int_C \frac{−(y+2)\,dx+(x−1)\,dy}{(x−1)^2+(y+2)^2}\), where \(C\) is any simple closed curve with an interior that does not contain point \((1,−2)\) traversed counterclockwise.
28. Evaluate \(\displaystyle \int_C \frac{x\,dx+y\,dy}{x^2+y^2}\), where \(C\) is any piecewise, smooth simple closed curve enclosing the origin, traversed counterclockwise.
- Answer
- \(\displaystyle \int_C \frac{x\,dx+y\,dy}{x^2+y^2}=2π\)
For the following exercises, use Green’s theorem to calculate the work done by force \(\vecs F\) on a particle that is moving counterclockwise around closed path \(C\) .
29. \(\vecs F(x,y)=xy\,\mathbf{\hat i}+(x+y)\,\mathbf{\hat j}, \quad C:x^2+y^2=4\)
30. \(\vecs F(x,y)=(x^{3/2}−3y)\,\mathbf{\hat i}+(6x+5\sqrt{y})\,\mathbf{\hat j}, \quad C\): boundary of a triangle with vertices \((0, 0), \,(5, 0),\) and \((0, 5)\)
- Answer
- \(W=\frac{225}{2}\) units of work
31. Evaluate \(\displaystyle \int_C (2x^3−y^3)\,dx+(x^3+y^3)\,dy\), where \(C\) is a unit circle oriented in the counterclockwise direction.
32. A particle starts at point \((−2,0)\), moves along the \(x\)-axis to \((2, 0)\), and then travels along semicircle \(y=\sqrt{4−x^2}\) to the starting point. Use Green’s theorem to find the work done on this particle by force field \(\vecs F(x,y)=x\,\mathbf{\hat i}+(x^3+3xy^2)\,\mathbf{\hat j}\).
- Answer
- \(W=12π\) units of work
33. David and Sandra are skating on a frictionless pond in the wind. David skates on the inside, going along a circle of radius \(2\) in a counterclockwise direction. Sandra skates once around a circle of radius \(3\), also in the counterclockwise direction. Suppose the force of the wind at point \((x,y)\) is \(\vecs F(x,y)=(x^2y+10y)\,\mathbf{\hat i}+(x^3+2xy^2)\,\mathbf{\hat j}\). Use Green’s theorem to determine who does more work.
34. Use Green’s theorem to find the work done by force field \(\vecs F(x,y)=(3y−4x)\,\mathbf{\hat i}+(4x−y)\,\mathbf{\hat j}\) when an object moves once counterclockwise around ellipse \(4x^2+y^2=4.\)
- Answer
- \(W=2π\) units of work
35. Use Green’s theorem to evaluate line integral \(\displaystyle ∮_C e^{2x}\sin 2y\,dx+e^{2x}\cos 2y\,dy\), where \(C\) is ellipse \(9(x−1)^2+4(y−3)^2=36\) oriented counterclockwise.
36. Evaluate line integral \(\displaystyle ∮_C y^2\,dx+x^2\,dy\), where \(C\) is the boundary of a triangle with vertices \((0,0), \,(1,1)\), and \((1,0)\), with the counterclockwise orientation.
- Answer
- \(\displaystyle ∮_C y^2\,dx+x^2\,dy=\frac{1}{3}\) units of work
37. Use Green’s theorem to evaluate line integral \(\displaystyle \int_C \vecs h·d\vecs r\) if \(\vecs h(x,y)=e^y\,\mathbf{\hat i}−\sin πx\,\mathbf{\hat j}\), where \(C\) is a triangle with vertices \((1, 0), \,(0, 1),\) and \((−1,0),\) traversed counterclockwise.
38. Use Green’s theorem to evaluate line integral \(\displaystyle \int_C\sqrt{1+x^3}\,dx+2xy\,dy\) where \(C\) is a triangle with vertices \((0, 0), \,(1, 0),\) and \((1, 3)\) oriented clockwise.
- Answer
- \(\displaystyle \int_C \sqrt{1+x^3}\,dx+2xy\,dy=3\) units of work
39. Use Green’s theorem to evaluate line integral \(\displaystyle \int_C x^2y\,dx−xy^2\,dy\) where \(C\) is a circle \(x^2+y^2=4\) oriented counterclockwise.
40. Use Green’s theorem to evaluate line integral \(\displaystyle \int_C \left(3y−e^{\sin x}\right)\,dx+\left(7x+\sqrt{y^4+1}\right)\,dy\) where \(C\) is circle \(x^2+y^2=9\) oriented in the counterclockwise direction.
- Answer
- \(\displaystyle \int_C \left(3y−e^{\sin x}\right)\,dx+\left(7x+\sqrt{y^4+1}\right)\,dy=36π\) units of work
41. Use Green’s theorem to evaluate line integral \(\displaystyle \int_C (3x−5y)\,dx+(x−6y)\,dy\), where \(C\) is ellipse \(\frac{x^2}{4}+y^2=1\) and is oriented in the counterclockwise direction.
42. Let \(C\) be a triangular closed curve from \((0, 0)\) to \((1, 0)\) to \((1, 1)\) and finally back to \((0, 0).\) Let \(\vecs F(x,y)=4y\,\mathbf{\hat i}+6x^2\,\mathbf{\hat j}.\) Use Green’s theorem to evaluate \(\displaystyle ∮_C\vecs F·d\vecs r.\)
- Answer
- \(\displaystyle ∮_C\vecs F·d\vecs r=2\) units of work
43. Use Green’s theorem to evaluate line integral \(\displaystyle ∮_C y\,dx−x\,dy\), where \(C\) is circle \(x^2+y^2=a^2\) oriented in the clockwise direction.
44. Use Green’s theorem to evaluate line integral \(\displaystyle ∮_C (y+x)\,dx+(x+\sin y)\,dy,\) where \(C\) is any smooth simple closed curve joining the origin to itself oriented in the counterclockwise direction.
- Answer
- \(\displaystyle ∮_C (y+x)\,dx+(x+\sin y)\,dy=0\) units of work
45. Use Green’s theorem to evaluate line integral \(\displaystyle ∮_C \left(y−\ln(x^2+y^2)\right)\,dx+\left(2\arctan \frac{y}{x}\right)\,dy,\) where \(C\) is the positively oriented circle \((x−2)^2+(y−3)^2=1.\)
46. Use Green’s theorem to evaluate \(\displaystyle ∮_C xy\,dx+x^3y^3\,dy,\) where \(C\) is a triangle with vertices \((0, 0), \,(1, 0),\) and \((1, 2)\) with positive orientation.
- Answer
- \(\displaystyle ∮_C xy\,dx+x^3y^3\,dy=2221\) units of work
47. Use Green’s theorem to evaluate line integral \(\displaystyle \int_C \sin y\,dx+x\cos y\,dy,\) where \(C\) is ellipse \(x^2+xy+y^2=1\) oriented in the counterclockwise direction.
48. Let \(\vecs F(x,y)=\left(\cos(x^5)−13y^3\right)\,\mathbf{\hat i}+13x^3\,\mathbf{\hat j}.\) Find the counterclockwise circulation \(\displaystyle ∮_C\vecs F·d\vecs r,\) where \(C\) is a curve consisting of the line segment joining \((−2,0)\) and \((−1,0),\) half circle \(y=\sqrt{1−x^2},\) the line segment joining \((1, 0)\) and \((2, 0),\) and half circle \(y=\sqrt{4−x^2}.\)
- Answer
- \(\displaystyle ∮_C\vecs F·d\vecs r=15π^4\) units of work
49. Use Green’s theorem to evaluate line integral \(\displaystyle ∫_C \sin(x^3)\,dx+2ye^{x^2}\,dy,\) where \(C\) is a triangular closed curve that connects the points \((0, 0), \,(2, 2),\) and \((0, 2)\) counterclockwise.
50. Let \(C\) be the boundary of square \(0≤x≤π,\;0≤y≤π,\) traversed counterclockwise. Use Green’s theorem to find \(\displaystyle ∫_C \sin(x+y)\,dx+\cos(x+y)\,dy.\)
- Answer
- \(\displaystyle \int_C\sin(x+y)\,dx+\cos(x+y)\,dy=4\) units of work
51. Use Green’s theorem to evaluate line integral \(\displaystyle ∫_C \vecs F·d\vecs r,\) where \(\vecs F(x,y)=(y^2−x^2)\,\mathbf{\hat i}+(x^2+y^2)\,\mathbf{\hat j},\) and \(C\) is a triangle bounded by \(y=0,\;x=3,\) and \(y=x,\) oriented counterclockwise.
52. Use Green’s Theorem to evaluate integral \(\displaystyle ∫_C \vecs F·d\vecs r,\) where \(\vecs F(x,y)=(xy^2)\,\mathbf{\hat i}+x\,\mathbf{\hat j},\) and \(C\) is a unit circle oriented in the counterclockwise direction.
- Answer
- \(\displaystyle ∫_C \vecs F·d\vecs r=π\) units of work
53. Use Green’s theorem in a plane to evaluate line integral \(\displaystyle ∮_C (xy+y^2)\,dx+x^2\,dy,\) where \(C\) is a closed curve of a region bounded by \(y=x\) and \(y=x^2\) oriented in the counterclockwise direction.
54. Calculate the outward flux of \(\vecs F(x,y)=−x\,\mathbf{\hat i}+2y\,\mathbf{\hat j}\) over a square with corners \((±1,\,±1),\) where the unit normal is outward pointing and oriented in the counterclockwise direction.
- Answer
- \(\displaystyle ∮_C\vecs F·\vecs N \,ds=4\)
55. [T] Let \(C\) be circle \(x^2+y^2=4\) oriented in the counterclockwise direction. Evaluate \(\displaystyle ∮_C \left[\left(3y−e^{\arctan x})\,dx+(7x+\sqrt{y^4+1}\right)\,dy\right]\) using a computer algebra system.
56. Find the flux of field \(\vecs F(x,y)=−x\,\mathbf{\hat i}+y\,\mathbf{\hat j}\) across \(x^2+y^2=16\) oriented in the counterclockwise direction.
- Answer
- \(\displaystyle ∮_C \vecs F·\vecs N\,ds=32π\)
57. Let \(\vecs F=(y^2−x^2)\,\mathbf{\hat i}+(x^2+y^2)\,\mathbf{\hat j},\) and let \(C\) be a triangle bounded by \(y=0, \,x=3,\) and \(y=x\) oriented in the counterclockwise direction. Find the outward flux of \(\vecs F\) through \(C\).
58. [T] Let \(C\) be unit circle \(x^2+y^2=1\) traversed once counterclockwise. Evaluate \(\displaystyle ∫_C \left[−y^3+\sin(xy)+xy\cos(xy)\right]\,dx+\left[x^3+x^2\cos(xy)\right]\,dy\) by using a computer algebra system.
- Answer
- \(\displaystyle ∫_C \left[−y^3+\sin(xy)+xy\cos(xy)\right]\,dx+\left[x^3+x^2\cos(xy)\right]\,dy=4.7124\) units of work
59. [T] Find the outward flux of vector field \(\vecs F(x,y)=xy^2\,\mathbf{\hat i}+x^2y\,\mathbf{\hat j}\) across the boundary of annulus \(R=\big\{(x,y):1≤x^2+y^2≤4\big\}=\big\{(r,θ):1≤r≤2,\,0≤θ≤2π\big\}\) using a computer algebra system.
60. Consider region \(R\) bounded by parabolas \(y=x^2\) and \(x=y^2.\) Let \(C\) be the boundary of \(R\) oriented counterclockwise. Use Green’s theorem to evaluate \(\displaystyle ∮_C \left(y+e^{\sqrt{x}}\right)\,dx+\left(2x+\cos(y^2)\right)\,dy.\)
- Answer
- \(\displaystyle ∮_C \left(y+e^{\sqrt{x}}\right)\,dx+\left(2x+\cos(y^2)\right)\,dy=13\) units of work
Contributors
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org .


1.1: Science and the Scientific Method
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Science is the process of describing the world around us. It is important to note that describing the world around us is not the same as explaining the world around us. Science aims to answer the question “How?” and not the question “Why?”. As we develop our description of the physical world, you should remember this important distinction and resist the urge to ask “Why?”.
The Scientific Method is a prescription for coming up with a description of the physical world that anyone can challenge and improve through performing experiments. If we come up with a description that can describe many observations, or the outcome of many different experiments, then we usually call that description a “Scientific Theory”. We can get some insight into the Scientific Method through a simple example.
Imagine that we wish to describe how long it takes for a tennis ball to reach the ground after being released from a certain height. One way to proceed is to describe how long it takes for a tennis ball to drop 1 m, and then to describe how long it takes for a tennis ball to drop 2 m, etc. We could generate a giant table showing how long it takes a tennis ball to drop from any given height. Someone would then be able to perform an experiment to measure how long a tennis ball takes to drop from 1 m or 2 m and see if their measurement disagrees with the tabulated values. If we collected the descriptions for all possible heights, then we would effectively have a valid and testable scientific theory that describes how long it takes tennis balls to drop from any height.
Suppose that a budding scientist, let’s call her Chloë, then came along and noticed that there is a pattern in the theory that can be described much more succinctly and generally than by using a giant table. In particular, suppose that she notices that, mathematically, the time, \(t\) , that it takes for a tennis ball to drop a height, \(h\) , is proportional to the square root of the height:
\[t \propto \sqrt{h}\]
Example \(\PageIndex{1}\)
Use Chloë’s Theory ( \(t \propto \sqrt{h}\) ) to determine how much longer it will take for an object to drop by \(2\) m than it would to drop by \(1\) m.
Solution
When we have a proportionality law (with a \(\propto\) ) sign, we can always change this to an equal sign by introducing a constant, which we will call \(k\) :
\[\begin{aligned} t &\propto \sqrt{h} \\[4pt] \rightarrow t&=k\sqrt{h}\end{aligned}\]
Let \(t_{1}\) be the time to fall a distance \(h_{1}=1\:\text{m}\) , and \(t_{2}\) be the time to fall a distance \(h_{2}=2\:\text{m}\) . In terms of our unknown constant, \(k\) , we have:
\[\begin{aligned} t_{1} &=k\sqrt{h_{1}}=k \sqrt{(1\:\text{m})}\\[4pt] t_{2} &=k\sqrt{h_{2}}=k \sqrt{(2\:\text{m})}\\[4pt]\end{aligned}\]
By taking the ratio, \(\frac{t_{1}}{t_{2}}\) , our unknown constant \(k\) will cancel:
\[\begin{aligned} \frac{t_{1}}{t_{2}}&=\frac{\sqrt{(1\:\text{m})}}{\sqrt{(2\:\text{m})}}=\frac{1}{\sqrt 2}\\[4pt] \therefore t_{2} &= \sqrt{2} t_{1}\end{aligned}\]
and we find that it will take \(\sqrt{2}\sim 1.41\) times longer to drop by \(2\) m than it will by \(1\) m.
Chloë’s “Theory of Tennis Ball Drop Times” is appealing because it is succinct, and it also allows us to make verifiable predictions . That is, using this theory, we can predict that it will take a tennis ball \(\sqrt 2\) times longer to drop from \(2\) m than it will from \(1\) m, and then perform an experiment to verify that prediction. If the experiment agrees with the prediction, then we conclude that Chloë’s theory adequately describes the result of that particular experiment. If the experiment does not agree with the prediction, then we conclude that the theory is not an adequate description of that experiment, and we try to find a new theory.
Chloë’s theory is also appealing because it can describe not only tennis balls, but the time it takes for other objects to fall as well. Scientists can then set out to continue testing her theory with a wide range of objects and drop heights to see if it describes those experiments as well. Inevitably, they will discover situations where Chloë’s theory fails to adequately describe the time that it takes for objects to fall (can you think of an example?).
We would then develop a new “Theory of Falling Objects” that would include Chloë’s theory that describes most objects falling, and additionally, a set of descriptions for the fall times for cases that are not described by Chloë’s theory. Ideally, we would seek a new theory that would also describe the new phenomena not described by Chloë’s theory in a succinct manner. There is of course no guarantee, ever, that such a theory would exist; it is just an optimistic hope of physicists to find the most general and succinct description of the physical world. This is a general difference between physics and many of the other sciences. In physics, one always tries to arrive at a succinct theory (e.g. an equation) that can describe many phenomena, whereas the other sciences are often very descriptive. For example, there is no succinct formula for how butterflies look; rather, there is a giant collection of observations of different butterflies.
This example highlights that applying the Scientific Method is an iterative process. Loosely, the prescription for applying the Scientific Method is:
- Identify and describe a process that is not currently described by a theory.
- Look at similar processes to see if they can be described in a similar way.
- Improve the description to arrive at a “Theory” that can be generalized to make predictions.
- Test predictions of the theory on new processes until a prediction fails.
- Improve the theory.
Exercise \(\PageIndex{1}\)
Fill in the blanks:
Physics is a branch of science that \(\underline{\qquad}\) the behavior of the universe. When doing physics, we attempt to answer the question of \(\underline{\qquad}\) things work the way they do.
- explains
- describes
- how
- why
- Answer
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B. describes
C. how


5.5: Margin of safety
The margin of safety looks at how far above the breakeven point a company’s sales are. The greater the difference, the more secure a company can feel about hedging against possible declines in sales. The margin of safety can be expressed as a dollar amount, a percentage, or a number of units.
As an example, a company’s breakeven point of 2,400 units per month is determined as follows:
\(\ \frac{\text { Total fixed costs }}{\text { Unit selling price - unit variable cost }}=\frac{\$ 120,000}{\$ 80-\$ 30}=2,400 \text{ units per month to break even}\)
Actual sales for the month were 8,000 units. The contribution margin per unit is $50 ($80 - $30).
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Margin of safety in units: |
8,000 – 2,400 = 5,600 |
|
Margin of safety in dollars: |
(8,000 x $50) – (2,400 x $50) = $400,000 - $120,000 = $280,000 |
|
Margin of safety percentage: |
($400,000 – 120,000) / $400,000 = 70% |
The margin of safety is 70%, which gives the company a significant cushion over its breakeven point. The higher the margin of safety, and the more it exceeds the breakeven point, the better.


1.1: Introduction to Identifying Word Parts
Learning Objectives
- Apply the basic rules for analyzing and building medical terms
- Identify four components of medical terms
- Define common prefixes, word roots, and suffixes
- Pronounce medical terms
- Apply the rules of medical language
- Identify meanings of key word components
Have you ever visited a health care professional and heard long, unfamiliar medical terms used to describe your condition? Many people feel as if medical professionals use a different language when talking about medical issues that can be difficult to understand. A medical term describes a word or condition that under normal circumstances would take several words to describe. Medical terminology facilitates effective and efficient communication by health care professionals by conveying a great quantity of information with precision in the fewest number of words.
This chapter will describe the components of medical terms, including common prefixes, word roots, and suffixes. By understanding the meaning of common components included in a medical term, you can easily construct the meaning of the term based on the meanings of its components. Specific medical terms related to body systems will be further discussed in the remaining chapters of this book.


4.1: Sketching Technique
Freehand sketching is a crucial skill that can be mastered with practice and a few fundamental guidelines. It's an essential tool for interpreting drawings and a valuable method for conveying job information to others, especially when no formal drawing is available. While professionally produced drawings may often be used in your work, there are many instances in the field where you might find yourself working off sketches drawn on various surfaces, from napkins and cardboard to wood scraps or any flat surface.
The ability to create technical sketches doesn't necessitate being an artist. Sketching isn't difficult if you follow a few simple rules, and while you might be slow at first, with practice, you'll be able to create reasonably good sketches without too much effort. The goal isn't to be judged as a professional draftsperson or architect but to be able to describe with lines what you need to be built, repaired, or modified. This requires practice and is an essential skill to develop.
For freehand sketching, a pad of graph paper (8 ½" × 11" sheets with a 5 mm or ¼" grid), a sharp HB pencil, and an eraser are the tools of the trade. Always ensure your pencil is sharp before starting any sketch, as this will enhance the quality and precision of your work. Remember, sometimes you'll need to work with what you have, using the tools and materials at hand, to communicate effectively about a project.
Sketching techniques
All you need to start with is a pencil and some paper. A soft pencil works best for most people, so try a #2 or an F. Keep the pencil sharp but not too sharp: hold it firmly enough for control but not so tight that your arm isn't relaxed. Don't draw heavily at first. That way, it is easier to erase without smudging. Darken the sketch when it begins to shape up the way you want it.
It's generally best to begin sketching with grid paper, although some prefer plain paper. On the job, you may find yourself sketching on the back of a work order or piece of packing crate. In any case, learning to sketch quickly and effectively is essential.
Here are some limbering-up exercises wot get you started. To keep your pencil sharper longer and for more precise lines widths, try turning your pencil slowly while completing the lines in the exercises below.
Sketching parallel lines
Start by drawing lines parallel to the paper's edges, such as a border line and title block. Use your finger as a guide when you draw along the grid line on the sketch pad (Figure \(\PageIndex{2}\)). Letting the end of your little finger run down the edge of the paper pad as you draw will steady your hand and make it easier to get a straight line.
Sketching non-parallel lines
When sketching lines that are not parallel to the sides of the paper, turn the paper around so that the line you wish to draw is either straight up and down in front of you or straight across the sheet of paper.
Drawing lines this way rather than at an angle across the sheet is much easier.  This will help you steady your hand (Figure \(\PageIndex{3}\)).
Sketching a rectangle
Locate the corners of the rectangle first. Then place your paper in a comfortable position for sketching and sketch downward for vertical lines and left to right for horizontal lines. Use the grid lines as a guide to help maintain your parallel and at 90 degrees lines to each other (Figure \(\PageIndex{4}\)).
Sketching a circle
First, locate the center of the circle (Figure \(\PageIndex{5}\)), and then very lightly box in the size of the circle (using the diameter as a guide), as in the top right. Sketch in the circle, one quarter at a time, as shown in the bottom row, left to right. You may find it necessary at first to add light points along the projected circumference to help guide you through each quarter. Remember to move your sketch pad to maintain a comfortable sketching position.
Sketching to approximate scale
Figure \(\PageIndex{6}\) shows the full square on the left. The center square is half size, and the right square is quarter size. Note that the center and right squares are the same shape as the left square, only smaller.
When sketching freehand, your sketches should reflect the actual shapes of objects as much as possible. Using grid paper, sketching to an approximate scale is not difficult. Assume that the object in Figure \(\PageIndex{7}\) is shown full size. As it is necessary to show all orthographic views on the same sheet of paper, the views must be scaled. Figure \(\PageIndex{8}\) shows the views at approximately one-half the original size.


2.1.10: Covered Structures for First-order Languages
Recall that a term is closed if it contains no variables.
If \(t\) is a closed term of the language \(\Lang L\) and \(\Struct M\) is a structure for \(\Lang L\) , the value \(\Value{t}{M}\) is defined as follows:
- If \(t\) is just the constant symbol \(c\) , then \(\Value{c}{M} = \Assign{c}{M}\) .
- If \(t\) is of the form \(\Atom{f}{t_1, \ldots, t_n}\) , then \[\Value{t}{M} = \Assign{f}{M}(\Value{t_1}{M}, \ldots, \Value{t_n}{M}).\nonumber\]
A structure is covered if every element of the domain is the value of some closed term.
Let \(\Lang L\) be the language with constant symbols \(\Obj{zero}\) , \(\Obj{one}\) , \(\Obj{two}\) , …, the binary predicate symbol \(<\) , and the binary function symbols \(+\) and \(\times\) . Then a structure \(\Struct M\) for \(\Lang L\) is the one with domain \(\Domain M = \{0, 1, 2, \ldots \}\) and assignments \(\Assign{\Obj{zero}}{M} = 0\) , \(\Assign{\Obj{one}}{M} = 1\) , \(\Assign{\Obj{two}}{M} = 2\) , and so forth. For the binary relation symbol \(<\) , the set \(\Assign{<}{M}\) is the set of all pairs \(\tuple{c_1, c_2} \in \Domain{M}^2\) such that \(c_1\) is less than \(c_2\) : for example, \(\tuple{1, 3} \in \Assign{<}{M}\) but \(\tuple{2, 2} \notin \Assign{<}{M}\) . For the binary function symbol \(+\) , define \(\Assign{+}{M}\) in the usual way—for example, \(\Assign{+}{M}(2,3)\) maps to \(5\) , and similarly for the binary function symbol \(\times\) . Hence, the value of \(\Obj{four}\) is just \(4\) , and the value of \(\times(\Obj{two}, +(\Obj{three},\Obj{zero}))\) (or in infix notation, \(\Obj{two} \times (\Obj{three} + \Obj{zero})\) ) is \[\begin{gathered} \begin{aligned} \Value{\times(\Obj{two}, +(\Obj{three},\Obj{zero})}{M} & =\Assign{\times}{M}(\Value{\Obj{two}}{M}, \Value{+(\Obj{three}, \Obj{zero})}{M})\\ & = \Assign{\times}{M}(\Value{\Obj{two}}{M}, \Assign{+}{M}(\Value{\Obj{three}}{M}, \Value{\Obj{zero}}{M})) \\ & = \Assign{\times}{M}(\Assign{\Obj{two}}{M}, \Assign{+}{M}(\Assign{\Obj{three}}{M}, \Assign{\Obj{zero}}{M})) \\ & = \Assign{\times}{M}(2, \Assign{+}{M}(3, 0)) \\ & = \Assign{\times}{M}(2, 3) \\ & = 6 \end{aligned}\end{gathered}\]
Is \(\Struct N\) , the standard model of arithmetic, covered? Explain.


11.5: White Nationalism
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- Erika Gutierrez, Janét Hund, Shaheen Johnson, Carlos Ramos, Lisette Rodriguez, & Joy Tsuhako
- Long Beach City College, Cerritos College, & Saddleback College via ASCCC Open Educational Resources Initiative (OERI)
"Free White Persons"
The Nationality Act of 1790 granted naturalization and citizenship in the United States to "free white persons" only and as we covered in Chapter 6.1 , immigration policies have kept the country largely white. However, by 2050, the U.S. is projected to become a majority-minority country, discussed further in Chapter 12.5 . Across U.S. history, some whites have met changes towards diversity and integration with violence and state legislation designed to restrict and prevent equal opportunity and advancement for all. In recent years, white nationalism (the belief that the United States should be a white ethnostate or white nation-state) has experienced a resurgence as populist attitudes, right-wing political beliefs, and anti-immigrant sentiment have been spurred on by an increasingly globalized world and changing racial landscape. White nationalists see themselves as protecting the Western world from non-white invasion and theft of resources and national identity which has shaped recent political campaigns in the U.S. that deploy "anti-globalist" and anti-immigrant rhetoric (Bonikowski & DiMaggio, 2016).
Radicalization Online
In 1995, there were only a few hate groups online; today there are hundreds. The internet is a low-cost and efficient medium with which to amplify the white nationalist message. According to hate group expert Mark Potok,
The internet is allowing the white supremacy movement to reach into places it has never reached before -- middle and upper middle-class, college bound teens. The movement is terribly interested in developing the leadership cadre of tomorrow...The movement is interested not so much in developing street thugs who beat up people in bars, but [in] college-bound teens who live in middle-class and upper-class homes (Swain, 2004).
Many of these websites present with informational resources where one can learn about American history and society. In fact, until recently one such site (stormfront.org) had a page dedicated to the "true historical examination" of Dr. Martin Luther King Jr. that depicted him as a communist, a drunk, and a rapist. The site also provided a link to download flyers which visitors were invited to distribute at their schools (Lee & Leets, 2002). Additionally, with the advent of social media (e.g., Facebook, Twitter, Instagram, etc.), radicalization has spiraled in the last several years. As explained in the 2020 documentary (see trailer below in Video 12.5.1), The Social Dilemma, technology via social media connects us but also controls us, divides us, monetizes us, manipulates us, polarizes us, distracts us, and divides us, to the point that former technology executives and designers predict a civil war as a result of the radicalization online.
Mainstreamers, Vanguardists & Alt-right
According to the Southern Poverty Law Center there are two major categories for pursuing a white enthnostate: mainstreaming and vanguardism . Mainstreamers seek to obtain power through an infiltration of traditional political institutions. The goal is to access positions that would put white nationalists in control of resources that could help to exclude and further marginalize non-whites such as instituting anti-immigration policies and eliminating social welfare programs.
Vanguardists take a more radical position that encourages a violent overthrow and seek to antagonize society towards a race-war and what they believe to be the inevitable collapse of America.
A third, more recent development merges these two styles and has come to be referred to as the "alt-right." Alt-right tactics focus on online activism in the form of "shitposting," meme making, and online harassment.
Unite the Right Rally
On the weekend of August 11, 2017, an estimated five hundred white supremacists and neo-Nazi's marched in the streets of Charlottesville, Virginia, decrying the proposed removal of a Robert E. Lee statue and chanting "Jews will not replace us!" and "Blood and soil!" They also chanted nationalist slogans representing Nazi Germany's ideal of a "racially" defined national body. Unlike the Ku Klux Klan which traditionally wears white robes and pointed hoods to conceal their identity, the participants in the "Unite the Right" rally donned tiki torches, white polo shirts and khaki slacks, uniforms that are linked to far-right groups such as Vanguard America and Identity Evropa, which has since rebranded as the American Identity Movement. The purpose of these strict fashion choices are to help the members distance themselves from their historical ideological roots and appear more mainstream and palatable to the broader public.
The following day, an estimated 1,000 counter-protestors, many of whom were ordinary residents of Charlottesville while others were part of more organized efforts from the faith-based groups, civil rights organizations, local businesses, and faculty and students at the University of Virginia, gathered to voice their disapproval of the rally. Tragically, one white protestor, Heather Heyer, 32, died when James Alex Fields Jr. plowed his car into a crowd of peaceful counter-protestors. In what some have argued is a lightly veiled approval of the rally's demands, President Trump remarked that "there were very fine people" on both sides of the protests spurring more debate on whether President Trump is a white nationalist himself, though he claimed he was merely voicing support for the defense of the Robert E. Lee statue (Kessler, 2020).
January 6, 2021
While most of the thousands of protesters were peaceful, an extreme incidence of violent domestic terrorism occurred on January 6, 2021 as alt-right, paramilitary and white supremacists stormed the U.S. Capitol building in Washington, D.C. Orchestrated by the Oath Keepers and Proud Boys, the violent protesters responded to the online, social media calls to "stop the steal," the (false) reference to President Biden stealing the election from former President Donald Trump - despite the fact that 7 million more Americans voted for Biden who also won the electoral college vote. Hundreds of individuals have been arrested for the violence in the Capitol, violence that was broadcast worldwide, including through the protesters' social media accounts, which in turn led to their tracking and arrests. The deaths of 5 individuals were attributed to the insurrection of January 6, and former President Trump was tried for impeachment in the House and Senate, though the latter fell short of the required 67 votes to impeach. National intelligence experts warn of the increasing threat of domestic violent terrorism, from both the political right and left, pose to our national security. Further, the fragility of our democracy was evidenced in this fateful event.
Contributors and Attributions
- Tsuhako, Joy. (Cerritos College)
- Johnson, Shaheen. (Long Beach City College)
References
- Bonikowski, B. & DiMaggio, P. (2016). Varieties of american popular nationalism. American Sociological Review 81 (5): 949-980.
- Kessler, G. (2020, May 8). The "very fine people" at charlottesville: who were they? The Washington Post.
- Lee, E. & Leets, L. (2002). Persuasive storytelling by hate groups online: examining its effects on adolescents, American Behavioral Scientist 45 , 927-957.
- McDermott, A. (1999, February 23). White hate group websites on the rise. CNN.
- Swain, C.M. (2004) The New white Nationalism in America: Its Challenge to Integration. Cambridge, UK: Cambridge University Press.


16.16: Articles
- Correctly identify and use indefinite and definite articles
Well, if you thought prepositions were “little” words, wait until we consider the part of speech called articles. Articles are similar to adjectives in that they modify nouns , but unlike adjectives, they don’t really describe a noun; they just identify a noun.
Articles are the smallest of the small but still serve an important function. We have three articles in the English language: a , an and the .
The is the definite article , which means it refers to a specific noun in a group.
A or an is the indefinite article , which means it refers to any member of a group. You would use the indefinite article when you aren’t trying to distinguish a particular noun.
The definite article indicates a level of specificity that the indefinite does not. “An apple” could refer to any apple; however “the apple” is referring to a specific apple. There are also cases in which no article is required:
- with generic nouns (plural or uncountable): cars have accelerators , happiness is contagious— which refer to cars and happiness in general (as compared with the happiness I felt yesterday , which specifies particular happiness)
- with most proper names: Sabrina , France , London , etc.
Indefinite Article
In English, the indefinite article takes the two forms a and an . These can be regarded as meaning “one,” usually without emphasis.
Distinction between a and an
You’ve probably learned the rule that an comes before a vowel, and that a comes before a consonant. While this is generally true, it’s more accurate to say that an comes before a vowel sound , and a comes before a consonant sound . Let’s look at a few examples with a :
- a box
- a HEPA filter (HEPA is pronounced as a word rather than as letters)
- a one-armed bandit (pronounced “won. . . “)
- a unicorn (pronounced “yoo. . . “)
And now with an :
- an apple
- an EPA policy (the letter E read as a letter still starts with a vowel sound)
- an SSO (pronounced “es-es-oh”)
- an hour (the h is silent)
- an heir (pronounced “air”)
Note: Some speakers and writers use an before a word beginning with the sound h in an unstressed syllable: an historical novel , an hotel . However, when the h is clearly pronounced, this usage is now less common, and a is preferred.
Definite Article
The definite article the is used when the referent of the noun phrase is assumed to be unique or known from the context. For example, in the sentence “ The boy with glasses was looking at the moon,” we can assume that, in this context, the reference is to only one boy and only one moon.
The can be used with both singular and plural nouns, with nouns of any gender, and with nouns that start with any letter. This is different from many other languages, which have different articles for different genders or numbers. The is the most commonly used word in the English language.
Word Order
In most cases, the article is the first word of its noun phrase, preceding all other adjectives and modifiers.
The little old red bag held a very big surprise.
There are a few exceptions, however:
- Certain determiners, such as all , both , half , and double , precede the definite article when used in combination ( all of the team , both of the girls , half the time , double the amount ).
 ).
- Adjectives qualified by too , so , as, and how generally precede the indefinite article: too great a loss , so hard a problem , as delicious an apple as I have ever tasted , I know how pretty a girl she is .
-
When adjectives are qualified by
quite
(particularly when it means “fairly”), the word
quite
(but not the adjective itself) often precedes the indefinite article:
quite a long letter
.
Note:
the phrase
a quite long letter
is also a correct construction. However the two have different meanings:
- In quite a long letter , quite modifies letter : it’s quite a letter.
- In a quite long letter , quite modifies long : the letter is quite long.
Read the following passage and make any necessary changes to the articles. Explain your reasoning for each change.
A Hubble Space Telescope (HST) is a space telescope that was launched into low Earth orbit in 1990 and remains in operation. Although not the first space telescope, Hubble is one of the largest and most versatile, and is well known as both an vital research tool and an public relations boon for astronomy. The HST is named after the astronomer Edwin Hubble.
Hubble’s orbit outside the distortion of Earth’s atmosphere allows it to take extremely high-resolution images. Hubble has recorded the some of most detailed visible-light images ever, allowing the deep view into space and time.
[practice-area rows=”4″][/practice-area]
[reveal-answer q=”666227″]Show Answer[/reveal-answer]
[hidden-answer a=”666227″]Here is the corrected passage. Each correction has been numbered. Explanations for each correction are given below the passage.
(1) The Hubble Space Telescope (HST) is a space telescope that was launched into low Earth orbit in 1990, and remains in operation. Although not the first space telescope, Hubble is one of the largest and most versatile, and is well known as both (2) a vital research tool and (3) a public relations boon for astronomy. The HST is named after the astronomer Edwin Hubble.
Hubble’s orbit outside the distortion of Earth’s atmosphere allows it to take extremely high-resolution images. Hubble has recorded (4) some of the most detailed visible-light images ever, allowing (5) a deep view into space and time.
Why were these changes necessary?
- There is only one Hubble Space Telescope, so it requires the definite article, the .
- Vital starts with a consonant sound, so it requires a, not an .
- Public starts with a consonant sound, so it requires a, not an .
- As we discussed, phrases like “some of” are exceptions to the general word order rule, and they come before articles.
- There are several different views into space, and this is just one of them. Thus, we need to use the indefinite article. Deep starts with a consonant, so it requires a, not an .
[/hidden-answer]
Contributors and Attributions
- Revision and Adaptation. Provided by : Lumen Learning. License : CC BY-SA: Attribution-ShareAlike
- Images of a, an, the. Provided by : Lumen Learning. License : CC BY: Attribution
- English articles. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/English_articles . License : CC BY-SA: Attribution-ShareAlike
- Modification of Hubble Space Telescope (with errors inserted). Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/Hubble_Space_Telescope . License : CC BY-SA: Attribution-ShareAlike
- Articles. Authored by : Excelsior OWL. Located at : https://owl.excelsior.edu/grammar-essentials/parts-of-speech/articles/ . License : CC BY: Attribution


6.2: Corporations and Stockholders’ Equity
A corporation is a form of business organization that is a separate legal entity ; it is distinct from the people who own it. The corporation can own property, enter into contracts, borrow money, conduct business, earn profit, pay taxes, and make investments similar to the way individuals can.
The owners of a corporation are called stockholders . These are people who have invested cash or contributed other assets to the business. In return, they receive shares of stock , which are transferable units of ownership in a corporation. Stock can also be thought of as a receipt to acknowledge ownership in the company. The value of the stock that a stockholder receives equals the value of the asset(s) that were contributed.
A corporation may be owned by one stockholder or by millions. Very small companies can incorporate by filing articles of incorporation with a state in the U.S. and being granted corporate status.
Corporations are ongoing . Stockholders can buy and sell their shares of stock without interrupting the operation of the company. Another characteristic of a corporation is limited liability . Stockholders can lose no more than the amount they invested in the corporation. If the corporation fails, the individuals who own it do not personally have to cover the corporation’s liabilities.
Up to this point, the stockholders’ equity section of the balance sheet has included two accounts: Common Stock and Retained Earnings . Common Stock is value that the owners have in the business because they have contributed their own personal assets. Retained earnings is value the owners have in the corporation because the business has been operating – doing what it was set up to do - and as a result it has generated a profit that the owners share. It is preferable, of course, for stockholder wealth to increase due to net income over time. That earnings potential is, in fact, what attracts stockholders to invest their own money into a business in the first place.
The following Accounts Summary Table summarizes the accounts relevant to issuing stock.
| ACCOUNT TYPE | ACCOUNTS | TO INCREASE | TO DECREASE | NORMAL BALANCE | FINANCIAL STATEMENT | CLOSE OUT? |
| Asset | Organization Costs | debit | credit | debit | Balance Sheet | NO |
| Liability | Cash Dividends Payable | credit | debit | credit | Balance Sheet | NO |
| Stockholders’ Equity |
Common Stock (CS)
|
credit | debit | credit | Balance Sheet | NO |
| Contra Stockholders’ Equity | Treasury Stock | debit | credit | debit | Balance Sheet | NO |
| Contra Stockholders’ Equity |
Cash Dividends
|
debit | credit | debit | Retained Earnings Statement | YES |
Common Stock , Preferred Stock , and Stock Dividends Distributable amounts can only be in multiples of par value. Use Paid-In Capital in Excess of Par for any differences between issue price and par value.
We will be using the accounts above in numerous journal entries. The point of these journal entries is to ultimately arrive at one number: total stockholders’ equity. Owners of a business are very interested in knowing what they are worth, and that final result is the answer to that question.
The new material we will cover next involves the stockholders’ equity section of the balance sheet. The generic Common Stock account will no longer be the only account used for owner investments: six new accounts will be added that describe a corporation’s equity in more specific detail. In addition, a second type of dividends will be covered: Stock Dividends .
The income statement is not affected by these new accounts. The retained earnings and balance sheet are. The statements on the left show account names in blue that you learned previously. The statements on the right show account names in blue that will replace those on the left as we take a more detailed look at stockholders’ equity.
The first five stockholders’ equity accounts shown on the balance sheet above track owner investments. The total value of these seven account balances is called paid-in capital . Total paid-in capital plus Retained Earnings , which is still used to keep a running balance of a company’s accumulated profit on hand, equals total stockholders’ equity.
Shares authorized is the number of shares a corporation is allowed to issue (sell). For a large corporation this is based on a decision by its Board of Directors, a group elected to represent and serve the interest of the stockholders. Authorization is just permission to sell shares of stock; no action has actually taken place yet. Therefore, there is no journal entry for a stock authorization.
Shares issued is the number of shares a corporation has sold to stockholders for the first time. The number of shares issued cannot exceed the number of shares authorized.
The terms above may be better understood with an analogy to a credit card. If you are approved for a credit card, the terms will include a credit limit, such as $5,000, which is the maximum that you are allowed to charge on the card. This is similar to “shares authorized,” the maximum number of shares a company is allowed to issue. The credit limit on a card does not mean that you have to charge $5,000 on your first purchase, but instead that you may continue to charge purchases up until you have reached a $5,000 maximum. The same holds true for shares issued. Smaller numbers of shares may be sold over time up to the maximum of the number of shares authorized.
If you wish to charge more than your credit limit on a credit card, you may contact the company that issued the card and request an increase in your credit limit. They may or may not grant this request. The same is true for a corporation. If it wishes to issue more shares than the number authorized, it may approach the Board of Directors with this request.


32: “Where is American Literature Now?”- Contemporary Connections Last updated Save as PDF Page ID 40686 Robin DeRosa, Abby Goode et al. Plymouth State University/ 32.1: Where is American Literature Now?- Nuclear War, Cyber-Attacks and Still Fighting With the Natives 32.2: Uncanny Parallels in Steven Universe 32.3: Does Twitter Poetry Count as American Literature? 32.4: American Literature- Helping Queer Kids Figure Themselves Out Since 1855 32.5: We are still making America, let alone making it great 32.6: I can do it…you can too!!! 32.7: The Tradition 32.8: It’s Always Sunny in America! 32.9: The (Super)Heroic (Time Travelers posing as) Slave(s) 32.10: The Theme of Rearticulation 32.11: Black Motherhood, The Desire For Family, and Black Heritage


16.2: Physiograhic Provinces of the United States of America
Module 16
Physiograhic Provinces of the United States of America
Introduction
The physiographic provinces of the United States of America can be broken into three different broad areas: Western, Central, and Eastern regions (Figure 1). Adjacent provinces will share features or will at least be a affected by the geologic events that define the nearby region. The Western Provinces are shaped by relatively young events (Post-Paleozoic), which are mostly the result of an active plate tectonic margin (the western edge of the continent is also the edge of a tectonic plate).
Figure 1. Geophysical Provinces of the Conterminous United States. Click to see larger.
The Eastern Provinces are a mix between young and old geologic events. Most of the individual provinces within this region result from the deformation of ancient mountain building as well as the recent passive sedimentary buildup observable today. Since the Central Provinces of the country has been largely shielded from tectonic activity, this area is flatter and less deformed than the coasts of the United States. Before the individual provinces are discussed in depth, it will be helpful to review the geologic time scale that was presented in earlier in this course.
Western U.S.
The geologic provinces in the Western United States occupy roughly a third of the country and stretch from the Rocky Mountains to the Pacific Ocean. We can group this tectonically active area into four types of provinces: the Rocky Mountains, the Coastal Pacific Mountain System, Interior Plateaus, and the extensional Basin and Range.
The Rocky Mountains are an immense range that stretches from New Mexico to Northern Canada. The Rocky Mountains first started to form during the Late Paleozoic, but their main growth occurred during the Laramide Orogeny during the Late Mesozoic and Early Cenozoic. An orogeny is a large-scale deformational event that is the result of the interaction between tectonic plates. In short, it is mountain building. The formation of this range differs from the Continent-Continent convergent boundaries, such as the Himalayas. The Rocky Mountains are the result of low-angle subduction, which caused deformation in the overlying plate. This striking mountain range is no longer active nor growing and has since been glaciated and eroded, which has rounded its highest peaks.
The Pacific Coast Mountains stretch from Mexico to the Arctic Circle in Alaska. Not only do the Pacific Coast Mountains differ significantly from the Rocky Mountains, but the mountain ranges within the Coastal Pacific Mountain System also differ amongst themselves. The Sierra Nevada Range runs 400 miles across the middle of California and contains iconic features such Yosemite’s Half Dome, which is a mecca for mountain climbers (Figure 2). As with the Rocky Mountains, the Sierra Nevada Range is the byproduct of convergence and subduction. During the same time as the growth of the Rocky Mountains (Late Mesozoic), the subduction led to the formation of volcanoes and the intrusion of massive granite batholiths. The volcanoes have long since eroded away exposing the batholiths at the surface. The Sierra Nevada topography is relatively recent (within the last 5 million years), as these rocks have been deformed by extensional stress, creating large fault blocks that build the steep face of the range. The Cascade Range is the northern portion of the Pacific Coast Mountain Range that spans an area from Northern California to Canada. The Cascade Range is the result of modern subduction of the Juan de Fuca and Gorda Plates under the North American Plate. Whereas the Sierra Nevada range is the eroded exposed core of a Mesozoic chain of volcanoes, the Cascade Range is modern and active. This chain encompasses thousands of volcanoes, including well-known landmarks such as Mount Rainier and Mount Saint Helens.
Figure 2. Half Dome at Yosemite National Park, which is part of the Sierra Nevada Range. Click to see larger.
Between the Rocky Mountains and the Pacific Coast Mountain Range is a large broad area containing a great diversity of geologic features. The Northernmost portion between the Cascades and the Rocky Mountains is the Columbia Plateau.  This hotspot produced several massive, though infrequent, eruptions; the largest eruption covering over 50,000 square miles. The Yellowstone Hotspot is still active and supplies the heat that feeds the classic geysers and hot springs throughout Yellowstone National Park (Figure 3). Southeast of this province is the Colorado Plateau, which is a broad expanse that, considering its neighboring provinces, is remarkably undeformed. The Colorado Plateau is mostly composed of horizontal Paleozoic and Mesozoic sedimentary rocks that contain abundant fossils, including the massive bone beds of Dinosaur National Monument. During the Laramide Orogeny this area was dramatically uplifted, allowing rivers to erode downward thus producing immense canyons, such as Grand Canyon National Park. The southeastern portion of the expanse between the Rocky Mountain and Pacific Coast Ranges is called the Basin and Range Province. This area can easily be recognized by the abundant horsts and grabens, which you may remember are series of elevated plateaus and low basins produced from the extensional deformation and normal faults. There are multiple hypotheses for the source of the extension, which range from heating from the mantle to movement along the San Andreas Fault.
Figure 3. Hot Spring in Yellowstone National Park within the Columbia Plateau Province. Click to see larger.
Eastern U.S.
The physiographic provinces that occupy the eastern third of the country range from New England to Texas, wrapping around the Gulf of Mexico. These provinces can easily be split into three regions: (1) provinces associated with the building of the Appalachian Mountains, (2) provinces built by much older orogenies in the Northeast, and (3) provinces created from more recent passive accumulation of sediments along the coast.
The Appalachian Mountains are the product of multiple collisions with small island chains during the Paleozoic, culminating with a collision with the supercontinent of Gondwana during the formation of Pangaea. These ancient mountains have been significantly eroded down to rolling hills that span most of the Southeastern United States as seen in Great Smoky Mountains National Park (Figure 4). The Appalachian Mountains can be further divided into four distinctive physiographic provinces based on their topography and geology. The center of the Appalachian Mountains is called the Valley and Ridge Province, which is composed of highly folded and faulted sedimentary rock. Fossils within these rocks indicate that they are Paleozoic in age, with thick and resistant Pennsylvanian-age sandstones forming its ridges while weaker Devonian and Cambrian shales form its valleys. To the east of the Valley and Ridge is the Blue Ridge Province, which was uplifted along with the Valley and Ridge but is composed of much older igneous and metamorphic rock. Absolute dating of these rocks indicates that the Blue Ridge was formed during the Proterozoic and later deformed during the Middle Paleozoic. The topography between these two provinces differs with the more uniform rocks in the Blue Ridge producing random peaks, valleys, and ridges as opposed to the resistant and non-resistant rocks of the Valley and Ridge, which produce more uniformly parallel ridges.
Figure 4. Appalachian Mountains within Great Smoky Mountains National Park, which contains multiple physiographic provinces. Click to see larger.
Southeast of the Blue Ridge is the Piedmont Province, which is typified by much lower, rolling hills along with small isolated mountains such as Stone Mountain in Georgia (Figure 5). The Piedmont is composed of igneous and metamorphic rock from the cores of long eroded mountain chains ranging from the Proterozoic to the end of the Paleozoic period. On the adjacent west side of the Valley and Ridge province is the Appalachian Plateau, which is an uplifted and largely undeformed region analogous to the Colorado Plateau. The rocks in this province are similar in age and lithology to those in the Valley and Ridge, but the preservation of fossils is enhanced because of the absence of extensive folding and faulting. In addition, the Appalachian Plateau contains abundant geological resources, including coal in West Virginia and iron in Alabama.
Figure 5. Stone Mountain within the Piedmont Province. Click to see larger.
The physiographic provinces in the northeastern United States are very similar to the Piedmont and Blue Ridge Provinces. The New England Province is similar to the Piedmont in that it is composed of Late Proterozoic and Paleozoic intrusive igneous and metamorphic rocks. The major difference between these two provinces is in their history following exposure, with the New England Province showing extensive weathering and erosion from glaciers that is absent from its southeastern counterpart. The Adirondack Mountain Province located in upstate New York is similar to the Blue Ridge in that they are both composed of igneous and metamorphic rock. However, the Adirondacks are fairly unique considering that they are a circular rather than linear range of mountains. The rocks themselves are billion-year-old remnants of the building of a Proterozoic supercontinent called Rodinia. The actual mountains in the range are quite young and represent Late Cenozoic uplift, which exposed these ancient rocks.
The last eastern physiographic province is the Coastal Plain that spans a vast area along the Atlantic Ocean and Gulf of Mexico from New England to Texas. This province is composed of the sediment that has accumulated since the rifting of Pangaea when the eastern edge of the continent became tectonically inactive. The sediment was derived from the continent and was deposited in shallow marine sedimentary environments with abundant, mature sediment and marine fossils.
Central U.S.
The central third of the country is mostly flat, undeformed, and dominated by sequences of sedimentary rocks. We separate this broad expanse into two physiographic provinces based largely on the source of the sediment. The Interior Lowlands Province covers the Midwestern states and consists largely of undeformed Paleozoic marine rocks (limestones and shales) that have been since carved and shaped by the activity of glaciers and rivers. This description may not seem as geologically exciting as those to the east or west, but spectacular cave developments can occur in these marine limestones, such as those from Mammoth Cave, Kentucky (Figure 6). To the west of the Interior Lowlands Province is the Great Plains Province, which is composed of sediments that eroded from the Rocky Mountains and are, therefore, substantially younger (Mesozoic and Cenozoic) with a higher proportion of clastic material. As with Interior Lowlands, the Great Plains have since been shaped by glaciers and rivers.
Figure 6. Mammoth Cave National Park in Central Kentucky. Click to see larger.


7.2: Later Etruscan Art
Etruscan Bronze Sculpture
Etruscan bronze classical art demonstrates the range and skill of artists who crafted large votive offerings and small, personal trinkets.
Discuss the different forms of bronze objects produced by the Etruscans
Key Points
- Few examples remain of any large-scale Etruscan bronze statues from the Classical era.
- The Chimera of Arezzo is a unique bronze votive offering to the god Tinia. The bronze sculpture demonstrates the Etruscans’ skill in bronze casting through the beast’s tense and plastic musculature, stylized mane, and fierce expression.
- The bronze Ombra della sera depicts the human body in an elongated fashion that is unusual for its time, while the Mars of Todi depicts a much more naturalistic representation. Small objects such as mirrors and cistae, small boxes, were also made of bronze during the Classical period. These objects were decorated with engravings or low relief casts that depict mythological scenes and were popular grave goods .
- While distinctly different from large-scale bronze, the small bronze objects demonstrate the variety and skill of Etruscan metal smiths that was first seen in the Orientalizing period with Etruscan jewelry and granulation techniques.
Key Terms
- engrave : To carve text or symbols into (something), usually for the purposes of identification or art.
- cista : A Latin word for a small box that used to hold unguents, jewels, or toiletries. They are generally used and related to women and domestic functions.
- pyxis : A small box.
Etruscan Bronze Sculptures
During its Classical period, from 480 to 300 BCE, Etruscan art was known for its hollow cast bronze sculptures and smaller utilitarian objects. Objects such as mirrors and cistae were engraved or incised with rich imagery that was made noticeable by a white substance inserted into the outlines and contour lines .
The Etruscan sculptures of human subjects range from naturalistic to exaggerated, almost modernist, forms , and the mythical creatures from the Greek and Eastern traditions found their way into their subject matter. In total, Etruscan sculpture incorporates a variety of cultural styles and motifs , including Orientalizing, Archaic , and Classical.
Very few Etruscan bronze sculptures survived because they were melted down in the following centuries to be used for other purposes.
Lost-Wax Casting
Also known by its French name cire perdue, lost-wax casting is the oldest method of producing metal sculptures. Although any material that can be liquified and subsequently solidified can be cast, metal is the most common. Of all metals, bronze is the most commonly cast.
In lost-wax casting, the sculptor begins by making a clay mold of the intended sculpture, coats it in wax, and applies an outer layer of plaster to it, fastening metal rods to hold the shell in place and wax rods to vent the mold. Intense heat is then applied, causing the wax to melt and flow out of the mold. This step leaves open channels into which molten metal is poured.
When the metal has cooled and hardened, the mold is broken, the sculpture is removed, and the holes left from the pins and rods are filled and smoothed. Finally, the sculpture is polished and ready for display. Because the mold must be destroyed, sculptors use the lost-wax method to produce one-of-a-kind sculptures.
Chimera of Arezzo
The bronze statue of the Chimera , discovered in the city of Arezzo in 1553, has recently been judged as a forgery. The reasoning rests more on the fact that there are no comparable existing Etruscan sculptures than on scientific testing.
An inscription on the right foreleg marks the bronze as a votive offering to the supreme Etruscan god, Tinia (comparable to Roman Jupiter). The sculpture depicts the Chimera, a mythical beast with the body and head of a lion, a serpent for a tail, and with a goat’s head in the middle of its back. The beast plagued the area of Lycia until it was slain by the hero Bellerophon.
The serpent tail was not discovered with the statue; the current tail is an eighteenth century restoration.
The bronze statue demonstrates Etruscan artistic techniques and how the liveliness seen in Etruscan terra cotta sculpture is transmitted into the bronze medium . The figure is well modeled and expressive—from its tense muscles, ready pose, and roaring face of the lion and goat. The beast’s ribs are clearly visible and a wound on its rump bleeds. The lion’s posture—he crouches and looks up—suggests that there may have been an additional figure of Bellerophon to create a votive group. The figure’s face and mane is stylized and the patterning is reminiscent of Near Eastern depictions of lions.
Ombra della sera
Produced in the metal-rich town of Velathri (later Volterra), the Ombra della sera (Italian for Evening Shadow) represents a male nude that is nearly two feet high. Archaeologists estimate its date of production to the third century BCE.
In ancient times, Volterra bronze workers were known for their ability to fuse copper ores with tin to form bronze. The body of the sculpture is very elongated, while the scale of its head is naturalistic.
The title Ombra della sera was applied to the sculpture centuries later by a poet who likened its exaggerated form to shadows cast by the setting sun. These unusual proportions lead some experts to believe that the sculpture was intended as a votive offering.
Mars of Todi
The so-called Mars of Todi (late fifth-early fourth century BCE) is a nearly life-size bronze warrior produced as a votive offering, possibly to Laran, the Etrucscan god of war. The figure probably held a patera (libation bowl) in his extended right hand, and a spear in the left.
His helmet is missing, but his body armor is one of the best surviving examples that shows how plate armor from the period appeared. Unlike the Ombra della sera, the Mars of Todi is very naturalistic, complete with a dynamic contrapposto pose and inlaid eyes.
A dedication that combines the Etruscan alphabet and Latin dialect from Umbria (central Italy) is inscribed on the skirt of the breastplate. It translates as, “Ahal Trutitis gave [this as a] gift.” Interestingly, the name of the donor was Celtic in origin, denoting the cosmopolitan nature of the region in ancient times.
Bronze Objects
The Etruscans also used bronze to create small objects, including storage jars, or cistae, and mirrors. Etruscan bronze mirrors were used by women and often deposited as grave goods. These mirrors were highly burnished on one side to reflect an image, and on the other side were decorated with engraved or low-relief casted scenes.
The images depicted were often scenes from Greek mythology, that demonstrate the importation of Greek culture and mythology. Many of these mirrors were inscribed with the deceased’s name, and ritual blessings.
Cistae were also often inscribed and were also decorated with engravings and added bronze elements, such as feet, chains, and decorative handle lids. The cistae were small boxes with lids, much like a pyxis , that was made from bronze and was usually cylindrical.
The handles of the lids were often figures, such as a man or a sphinx , or figural groups. Like the mirrors, the engraved scenes represented images from mythology, but some images also depict scenes from Etruscan history.
Etruscan Art under the Influence of the Romans
As the Etruscan territories fell under Roman dominance, Etruscan culture and art changed to reflect Roman influence and new cultural values.
Describe the changes that appeared in Etruscan art during the Roman period
Key Points
- Funerary imagery changed in Etruscan society as the Etruscans began to encounter violence and conquest by their Roman neighbors. The figures of the underworld demons Charun and Vanth, as well as apotropaic figures, are depicted in tomb paintings and and on sarcophagi during the Etrusco-Roman period.
- The Sarcophagus of Lars Pulena not only depicts the figures of Charun and Vanth, the lid also depicts the deceased as a prosperous and wealthy Etruscan, complete with a scroll listing his accomplishments. The sarcophagus demonstrates the shifting values in Etruscan society.
- The life-size bronze statue of Aule Metele depicts an Etruscan man dressed and behaving as a Roman. The man wears a toga and Roman sandals and is posed in a position of oration, as would be common for a Roman magistrate. The sculpture demonstrates the mixing of Etruscan and Roman styles and culture .
- Temple sculpture also changed during the Roman period, with terra cotta pedimental sculptures adorning at least two temples during the second century BCE. Both sculpture groups display Graeco-Roman naturalism in the depiction of the body. One group wears Roman-style military attire.
Key Terms
- veristic : Realistic; true to life.
- cinerary urn : A vessel used to hold the ash remains of the cremated deceased.
Etruscan Art under Rome
In 509 BCE, the Etruscan kings of Rome were expelled from the city, and the Roman Republic was established. By the fourth century BCE, Rome was beginning to expand across the Italian peninsula, and the first Etruscan city to fall was Veii in 396 BCE.
Over the following centuries, Etruria was involved in Roman wars, and Etruscan territory was fully conquered by the Romans by the beginning of the first century BCE. While Roman culture drew from its Etruscan roots, borrowing and adapting Etruscan customs, Etruscan society was also influenced by Roman culture.
During this period, art begin to adopt a Roman style and display the permeation of Roman culture and values into Etruscan society. The threat of invasion also led to the common presence of violence, especially in funerary images.
Funerary Art and Sarcophagi
Funerary art , both in tomb paintings and on carved sarcophagi, underwent a noticeable change in subject matter during the Roman period. The figures of Charun and Vanth, demons of the underworld, were depicted with increasing regularity.
Both figures are often depicted with wings, while Charun is often depicted with blue skin to signify putrefying flesh. They also carry torches, used to light the way to the underworld, or sometimes keys to open the door to the underworld, which underline the figures’ role as guides between the world of the living and the world of the dead. In the tomb paintings in Tarquinia, the figures of Charun and Vanth can be seen painted in front of or around doorways.
While Charun’s name is likely a derivative of the Greek underworld ferryman Charon, Vanth appears to be uniquely Etruscan. Due to Charun’s menacing associations, theories have attempted to associate Vanth with the avenging Greek Furies. However, her role as a benevolent guide conflicts with this suggestion.
Regardless of Vanth’s exact role and origins, the appearance of a less-than-joyous afterlife and menacing figures in Etruscan funerary art does not emerge until after the beginning of the Roman incursions into Etruscan territory. Perhaps Vanth is a gentler apotropaic figure, offering the reassurance of an ally in the afterlife to counteract the trials faced in the face of impending cultural collapse and absorption.
The Sarcophagus of Lars Pulena
Charun and Vanth also appear on stone and terra cotta sarcophagi. Charun is also sometimes depicted with a hammer. On the Sarcophagus of Lars Pulena, two figures of Charun (with hammers but without wings) are depicted on either side of a central figure, most likely Lars Pulena, swinging their hammers at his head. The violent image might have been used as an apotropaic device to ward off evil.
However, in comparison to earlier funerary images, the level of violence seems to mimic the new level of violence in Etruscan society from Roman forces and influence. Two winged representations of Vanth also appear on the sarcophagus, at either end of the frieze .
The lid of the sarcophagus depicts a portrait of the deceased. The man lies alone, wearing a somber expression, unlike the earlier terra cotta Sacophagus of the Spouses.
His face is wrinkled and reflects a Roman republican portrait style, which equates age with wisdom and leadership capabilities. He has a pot belly, signifying his wealth, good life, and robust eating, and he holds a scroll across his lap that is inscribed with a list of his accomplishments.
Smaller cinerary urns assumed the shapes of sarcophagi during this period. These urns are topped with images of the deceased lying across the lid, often in Roman dress, with relief-carved scenes of battle, violence, or Charun and Vanth.
The woman who reclines atop the urn wears attire more akin to that of a Roman matron than to the woman on the Sarcophagus of the Spouses. Unlike the Etruscans, who buried their dead in tombs designed to mimic the appearance and comforts of private homes, the Romans practiced cremation and stored the ashes of their deceased in cinerary urns. This shift in Etruscan culture demonstrates the adoption of Roman funerary practices.
Aule Metele
Aule Metele, also known as The Orator, is a life-size bronze sculpture of an Etrusco-Roman man. The figure is depicted wearing a Roman toga and Roman sandals. He stands in a pose of an orator, with his hand raised to address a crowd. To further emphasize this gesture, the hand is slightly enlarged.
He is clearly depicted as an individual, and an inscription on the hem of his toga in Etruscan names him as Aule Metele. Aule Metele dresses as a Roman magistrate, and his face is a cross between Hellenistic and Roman veristic portraiture.
The sculpture shows a level of individuality through the gaunt cheeks, thin lips, and wrinkled forehead. As with the sculpture on the Sarcophagus of Lars Pulena, these attributes of age align with the respect afforded to elders in Roman society. While the inscription marks him as an Etruscan, his attire and pose demonstrate the absorption of Roman culture into Etruscan society and the adoption—especially by the ruling class of Etruscans—of Roman civic practices.
Late Temple Sculpture
By the second century BCE, at least two Etruscan temples began to show evidence of the absorption of the Roman culture. Unlike early temples, whose pediments were largely unadorned, the temples in the cities of Luna and Talamone incorporated pedimental sculptures in the style of the Greeks.
Although Roman pediments remained free of sculpture groups, the Roman influence is clearly visible in these terra cotta figures. Whereas pre-Roman temple sculptures were largely stylized like the Apulu of Veii, the pedimental figures from the temples at Luna and Talamone possess the naturalism of Classical and Hellenistic sculpture, both of which were adopted by the Romans.
Unlike Greek pedimental sculpture that depicted male nudes, the figures from the pediment from Talamone, which depicts the fate of the Seven against Thebes, wear Roman battle gear, including short-sleeved skirted armor. By the time the sculpture groups from both temples were produced, their cities were under Roman control.
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