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Chapter 14: Problem 6

What does it mean if the curl of a vector field is zero throughout a region?

Short Answer

Expert verified
Answer: When the curl of a vector field is zero throughout a region, it means that the vector field does not exhibit any local rotation or circulation in that region. The field is conservative, and its circulation around any closed loop within that region is also zero. This implies that there is a potential function associated with the vector field, which allows us to determine the work done by the field when moving from one point to another without knowing the path taken.

Step by step solution

01

Understanding curl of a vector field

In vector calculus, curl is a measure of the rotation or the circulation present in a given vector field. Specifically, curl is a vector field that represents the local rotation or the vorticity at a given point in the vector field. Mathematically, it is represented as the cross product of the nabla operator (鈭�) and the vector field (F), i.e., curl(F) = 鈭� 脳 F.
02

Connection to the circulation of the vector field

Circulation refers to the integral of a vector field along a closed curve within the given region. It measures the tendency of the vector field to circulate around points in the region. If the curl of a vector field is nonzero, that means there is some rotation or circulation present at that point, and the vector field is not conservative. However, if the curl is zero, it implies that the vector field is curl-free or irrotational.
03

Interpreting curl of zero throughout a region

If the curl of a vector field is zero throughout a region, it implies that the vector field does not exhibit any local rotation or circulation in that region. In other words, the field is conservative, and its circulation around any closed loop within that region is also zero. Additionally, a conservative vector field (with a curl of zero) has a potential function associated with it. This potential function allows us to determine the work done by the field when moving from one point to another without knowing the path taken, since the work done only depends on the difference in the potential function's values at those two points.

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Most popular questions from this chapter

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 \(z \geq 0\).

a. Prove that the rotation field \(\mathbf{F}=\frac{\langle-y, x\rangle}{|\mathbf{r}|^{p}},\) where \(\mathbf{r}=\langle x, y\rangle\) is not conservative for \(p \neq 2\) b. For \(p=2,\) show that \(\mathbf{F}\) is conservative on any region not containing the origin. c. Find a potential function for \(\mathbf{F}\) when \(p=2\)

Suppose \(y=f(x)\) is a continuous and positive function on \([a, b] .\) Let \(S\) be the surface generated when the graph of \(f\) on \([a, b]\) is revolved about the \(x\) -axis. a. Show that \(S\) is described parametrically by \(\mathbf{r}(u, v)=\langle u, f(u) \cos v, f(u) \sin v\rangle,\) for \(a \leq u \leq b, 0 \leq v \leq 2 \pi\) b. Find an integral that gives the surface area of \(S\) c. Apply the result of part (b) to find the area of the surface generated with \(f(x)=x^{3},\) for \(1 \leq x \leq 2\) d. Apply the result of part (b) to find the area of the surface generated with \(f(x)=\left(25-x^{2}\right)^{1 / 2},\) for \(3 \leq x \leq 4\).

Prove that the radial field \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}},\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(p\) is a real number, is conservative on any region not containing the origin. For what values of \(p\) is \(\mathbf{F}\) conservative on a region that contains the origin?

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. What is the length of \(C ?\)
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