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Chapter 14: Problem 1

Explain the meaning of the integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) in Stokes' Theorem.

Short Answer

Expert verified
Answer: In the context of Stokes' Theorem, the integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) represents the circulation of the vector field \(\mathbf{F}\) around a closed curve \(C\), quantifying the twisting or rotational aspects of the vector field. According to Stokes' Theorem, this circulation is equal to the surface integral of the curl of the vector field over the surface bounded by the curve.

Step by step solution

01

Define the line integral

The line integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) is the integral of the dot product of a vector field \(\mathbf{F}\) and a differential displacement vector \(d \mathbf{r}\) along a closed curve \(C\). It represents the work done by the vector field \(\mathbf{F}\) along the curve \(C\).
02

Introduce Stokes' Theorem

Stokes' Theorem relates the line integral of a vector field \(\mathbf{F}\) around a closed curve \(C\) to the surface integral of the curl of the vector field over the smooth surface \(S\) bounded by the curve \(C\). Mathematically, the theorem is expressed as: $$\oint_{C} \mathbf{F} \cdot d \mathbf{r} = \iint_{S} \nabla \times \mathbf{F} \cdot d \mathbf{S}$$
03

Explain the meaning of the line integral in Stokes' Theorem

In the context of Stokes' Theorem, the line integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) represents the circulation of the vector field \(\mathbf{F}\) around the closed curve \(C\). Circulation can be understood as a measure of how much the vector field twists around the curve. If the circulation is zero, it means that the vector field is conservative (i.e., it has no rotational components) and the work done by the vector field around the curve is zero.
04

Relate the line integral to the surface integral

Stokes' Theorem states that the circulation of a vector field \(\mathbf{F}\) around a closed curve \(C\) is equal to the surface integral of the curl of the vector field over the smooth surface \(S\) bounded by the curve \(C\). This relation implies that, in regions where the curl of the vector field is significant, there is more likely to be a significant circulation around the curve \(C\). In other words, the line integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) quantifies the swirling or rotational properties of the vector field around the curve. In summary, the integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) in Stokes' Theorem represents the circulation of the vector field \(\mathbf{F}\) around a closed curve \(C\), quantifying the twisting or rotational aspects of the vector field. According to Stokes' Theorem, this circulation is equal to the surface integral of the curl of the vector field over the surface bounded by the curve.

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Most popular questions from this chapter

Prove the following properties of the divergence and curl. Assume \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields and \(c\) is a real number. a. \(\nabla \cdot(\mathbf{F}+\mathbf{G})=\nabla \cdot \mathbf{F}+\nabla \cdot \mathbf{G}\) b. \(\nabla \times(\mathbf{F}+\mathbf{G})=(\nabla \times \mathbf{F})+(\nabla \times \mathbf{G})\) c. \(\nabla \cdot(c \mathbf{F})=c(\nabla \cdot \mathbf{F})\) d. \(\nabla \times(c \mathbf{F})=c(\nabla \times \mathbf{F})\)

The cone \(z^{2}=x^{2}+y^{2},\) for \(z \geq 0,\) cuts the sphere \(x^{2}+y^{2}+z^{2}=16\) along a curve \(C\) a. Find the surface area of the sphere below \(C,\) for \(z \geq 0\). b. Find the surface area of the sphere above \(C\). c. Find the surface area of the cone below \(C,\) for \(z \geq 0\).

Let S be the disk enclosed by the curve \(C: \mathbf{r}(t)=\langle\cos \varphi \cos t, \sin t, \sin \varphi \cos t\rangle,\)for \(0 \leq t \leq 2 \pi,\) where \(0 \leq \varphi \leq \pi / 2\) is a fixed angle. Use Stokes' Theorem and a surface integral to find the circulation on \(C\) of the vector field \(\mathbf{F}=\langle-y, x, 0\rangle\) as a function of \(\varphi .\) For what value of \(\varphi\) is the circulation a maximum?

Let \(S\) be the cylinder \(x^{2}+y^{2}=a^{2},\) for \(-L \leq z \leq L\) a. Find the outward flux of the field \(\mathbf{F}=\langle x, y, 0\rangle\) across \(S\) b. Find the outward flux of the field \(\mathbf{F}=\frac{\langle x, y, 0\rangle}{\left(x^{2}+y^{2}\right)^{p / 2}}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}}\) across \(S,\) where \(|\mathbf{r}|\) is the distance from the \(z\) -axis and \(p\) is a real number. c. In part (b), for what values of \(p\) is the outward flux finite as \(a \rightarrow \infty\) (with \(L\) fixed)? d. In part (b), for what values of \(p\) is the outward flux finite as \(L \rightarrow \infty\) (with \(a\) fixed)?

Suppose a solid object in \(\mathbb{R}^{3}\) has a temperature distribution given by \(T(x, y, z) .\) The heat flow vector field in the object is \(\mathbf{F}=-k \nabla T,\) where the conductivity \(k>0\) is a property of the material. Note that the heat flow vector points in the direction opposite that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is \(\nabla \cdot \mathbf{F}=-k \nabla \cdot \nabla T=-k \nabla^{2} T\) (the Laplacian of \(T\)). Compute the heat flow vector field and its divergence for the following temperature distributions. $$T(x, y, z)=100(1+\sqrt{x^{2}+y^{2}+z^{2}})$$
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