Chapter 15

Suppose that a surface \(\sigma\) in 3 -space and a function \(f(x, y, z)\) are described geometrically. For example, \(\sigma\) might be the sphere of radius 1 centered at the origin and \(f(x, y, z)\) might be the distance from the point \((x, y, z)\) to the \(z\) -axis. How would you explain to a classmate a procedure for evaluating the surface integral of \(f\) over \(\sigma ?\)

Writing Use the Internet or other sources to find information about "planimeters," and then write a paragraph that describes the relationship between these devices and Green's Theorem.

Verify that the radius vector r = xi + y j + zk has the stated property. $$ \text { (a) div } \mathbf{r}=3 \quad \text { (b) } \nabla \frac{1}{\|\mathbf{r}\|}=-\frac{\mathbf{r}}{\|\mathbf{r}\|^{3}} $$

Find the work done by the force field \(\mathbf{F}\) on a particle that moves along the curve \(C\). $$ \begin{array}{l}{\mathbf{F}(x, y)=\left(x^{2}+x y\right) \mathbf{i}+\left(y-x^{2} y\right) \mathbf{j}} \\ {C: x=t, \quad y=1 / t \quad(1 \leq t \leq 3)}\end{array} $$

Find the work done by the force field \(\mathbf{F}\) on a particle that moves along the curve \(C\). $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}+x z \mathbf{k}} \\ {C: \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k} \quad(0 \leq t \leq 1)}\end{array} $$

Find the work done by the force field \(\mathbf{F}\) on a particle that moves along the curve \(C\). \(\mathbf{F}(x, y, z)=(x+y) \mathbf{i}+x y \mathbf{j}-z^{2} \mathbf{k}\) \(C:\) along line segments from \((0,0,0)\) to \((1,3,1)\) to \((2,-1,4)\)

A curve \(C\) is called a flow line of a vector field \(\mathbf{F}\) if \(\mathbf{F}\) is a tangent vector to \(C\) at each point along \(C\) (see the accom- panying figure). (a) Let \(C\) be a flow line for \(\mathbf{F}(x, y)=-y \mathbf{i}+x \mathbf{j},\) and let \((x, y)\) be a point on \(C\) for which \(y \neq 0 .\) Show that the flow lines satisfy the differential equation $$ \frac{d y}{d x}=-\frac{x}{y} $$ Solve the differential equation in part (a) by separation of variables, and show that the flow lines are concentric circles centered at the origin.

Use a line integral to find the area of the surface. The surface that extends upward from the parabola \(y=x^{2}(0 \leq x \leq 2)\) in the \(x y\) -plane to the plane \(z=3 x\)

Evaluate the integral \(\int_{-C} \frac{x d y-y d x}{x^{2}+y^{2}},\) where \(C\) is the circle \(x^{2}+y^{2}=a^{2}\) traversed counterclockwise.

Suppose that a particle moves through the force field \(\mathbf{F}(x, y)=x y \mathbf{i}+(x-y) \mathbf{j}\) from the point \((0,0)\) to the point \((1,0)\) along the curve \(x=t, y=\lambda t(1-t) .\) For what value of \(\lambda\) will the work done by the force field be \(1 ?\)

Writing In physical applications it is often necessary to deal with vector quantities that depend not only on position in space but also on time. Give some examples and discuss how the concept of a vector field would need to be modified to apply to such situations.

A farmer weighing 150 lb carries a sack of grain weighing 20 lb up a circular helical staircase around a silo of radius \(25 \mathrm{ft}\). As the farmer climbs, grain leaks from the sack at a rate of 1 lb per \(10 \mathrm{ft}\) of ascent. How much work is performed by the farmer in climbing through a vertical distance of \(60 \mathrm{ft}\) in exactly four revolutions? [Hint: Find a vector field that represents the force exerted by the farmer in lifting his own weight plus the weight of the sack upward at each point along his path.]

Describe the different types of line integrals, and discuss how they are related.