Chapter 15

Find curl \(\mathbf{F}\). $$\mathbf{F}=z \mathbf{i}+x j+y \mathbf{k}$$

Compute the flux \(\iint \mathbf{F} \cdot \mathbf{n} d S\) by the Divergence Theorem. \(\mathbf{F}=x \mathbf{i}+x \mathbf{j}+x \mathbf{k}\) S: unit sphere \(x^{2}+y^{2}+z^{2}-1\)

Compute the line integrals. \(\int_{c} d s\) and \(\int_{c} d y: x=t, y=2 t, 0 \leqslant t \leqslant 1\).

Find curl \(\mathbf{F}\). $$\mathbf{F}=\operatorname{grad}\left(x e^{y} \sin z\right)$$

Compute the flux \(\iint \mathbf{F} \cdot \mathbf{n} d S\) by the Divergence Theorem. \(F=-y \mathbf{i}+x \mathbf{j} . \quad V:\) unit cube \(0 \leqslant x \leqslant 1,0 \leqslant y \leqslant 1,0 \leqslant z \leqslant 1\)

Find a potential \(f(x, y)\) for the gradient fields \(1-8\). Draw the streamlines perpendicular to the equipotentials \(f(x, y)=c .\) $$\mathbf{F}=\mathbf{x i}+\mathbf{j}$$

Compute the flux \(\iint \mathbf{F} \cdot \mathbf{n} d S\) by the Divergence Theorem. \(\mathbf{F}=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k}, \quad S:\) unit sphere

Find curl \(\mathbf{F}\). $$\mathbf{F}=(x+y) \mathbf{i}-(x+y) \mathbf{k}$$

Compute the flux \(\iint \mathbf{F} \cdot \mathbf{n} d S\) by the Divergence Theorem. \(\mathbf{F}=x^{2} \mathbf{i}+8 \mathbf{y}^{2} \mathbf{j}+\boldsymbol{z}^{2} \mathbf{k}, \quad V ;\) unit cube.

Find a potential \(f(x, y)\) for the gradient fields \(1-8\). Draw the streamlines perpendicular to the equipotentials \(f(x, y)=c .\) $$\mathbf{F}=(1 / y) \mathbf{i}-\left(x / y^{2}\right) \mathbf{j}$$

Compute the line integrals. \(\int c d x\) and \(\int_{c} y d x:\) any closed circte of radius \(3 .\)

Compute the flux \(\iint \mathbf{F} \cdot \mathbf{n} d S\) by the Divergence Theorem. \(\mathbf{F}=\mathbf{u}_{r}=(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}) \rho, \quad S:\) sphere \(p=a\)

Compute the line integrals. \(\int_{c}(d s / d t) d t:\) any path of length \(5 .\)

Compute the flux \(\iint \mathbf{F} \cdot \mathbf{n} d S\) by the Divergence Theorem. \(\mathbf{F}=x^{3} \mathbf{i}+y^{3} \mathbf{j}+z^{3} \mathbf{k}, \quad\) S: sphere \(\rho=a\)

Show that \(\oint_{c}\left(x^{2} y+2 x\right) d y+x y^{2} d x\) depends only on the area of \(R\). Does it equal the area?

Does \(\int_{p}^{Q} x d x\) equal \(\left.\frac{1}{2} x^{2}\right]_{P}^{Q} ?\)

Compute the flux \(\iint \mathbf{F} \cdot \mathbf{n} d S\) by the Divergence Theorem. \(\mathbf{F}=z^{2} \mathbf{k}, \quad V:\) upper half of ball \(\rho \leqslant a\)

Draw the shear field \(F=x j\). Check that it is not a gradient field: If \(\partial f / \partial x=0\) then \(\partial f / \partial y=x\) is impossible. What are the streamlines (field lines) in the direction of \(\mathbf{F} ?\)

Find \(N\) and \(d S=|\mathbf{N}| d x d y\) and the surface area \(\iint d S\) Integrate over the \(x y\) shadow which ends where the \(z\) 's are equal \(\left(x^{2}+y^{2}=4\right.\) in Problem 1). \(z=x+y\) above triangle with vertices (0,0),(2,2),(0,2).

Find all functions that satisfy \(\partial f / \partial x=-y\) and show that none of them satisfy \(\partial f / \partial y=x\). Then the spin field \(S=\) \(-y \mathbf{i}+x \mathbf{j}\) is not a gradient field.

Find \(\iiint \operatorname{div}\left(x^{2} \mathbf{i}+y \mathbf{j}+2 \mathbf{k}\right) d V\) in the cube \(0 \leqslant x, y, z \leqslant a\) Also compute \(\mathbf{n}\) and \(\iint \mathbf{F} \cdot \mathbf{n} d S\) for all six faces and add.

Find the work in moving from (1,0) to \\{0,1) . When \(F\) is conservative, construct \(f\). Choose yout own path when \(\mathbf{F}\) is not eonservative. $$\mathbf{F}=\mathbf{i}+y \mathbf{j}$$

Compute curl \(F\) and find \(\oint_{C} F \cdot d R\) by Stokes? Theorem. \(\mathbf{F}=\mathbf{i} \times \mathbf{R}, C=\) circle \(x^{2}+z^{2}=1, y=0\).

Devise a way to find the one-dimensional theorem \(\int_{a}^{b}(d f / d x) d x=f(b)-f(a)\) as a special case of Green's Theorem when \(R\) is a square.

Compute \(\partial f / \partial x\) and \(\partial f / \partial y\) in \(11-18\). Draw the gradient field \(\mathbf{F}=\) grad \(f\) and the equipotentials \(f(x, y)=c:\) $$f=x-3 y$$

Compute curl \(F\) and find \(\oint_{C} F \cdot d R\) by Stokes? Theorem. \(\mathbf{F}=(\mathbf{i}+\mathbf{i}) \times \mathbf{R} \cdot C=\operatorname{circle} v^{2}+z^{2}=1, x=0\).

Find the work in moving from (1,0) to \\{0,1) . When \(F\) is conservative, construct \(f\). Choose yout own path when \(\mathbf{F}\) is not eonservative. $$\mathbf{F}=e^{y} \mathbf{i}+x e^{y} \mathbf{j}$$

Compute \(\partial f / \partial x\) and \(\partial f / \partial y\) in \(11-18\). Draw the gradient field \(\mathbf{F}=\) grad \(f\) and the equipotentials \(f(x, y)=c:\) $$f=(x-1)^{2}+y^{2}$$

Compute \(\partial f / \partial x\) and \(\partial f / \partial y\) in \(11-18\). Draw the gradient field \(\mathbf{F}=\) grad \(f\) and the equipotentials \(f(x, y)=c:\) $$f=x^{2}-y^{2}$$

Compute the surface integrals \(\iint g(x, y, z) d S\). \(g=x^{2}+y^{2}\) over the top half of \(x^{2}+y^{2}+z^{2}=1(\) use \(\phi, \theta)\).

Find the work in moving from (1,0) to \\{0,1) . When \(F\) is conservative, construct \(f\). Choose yout own path when \(\mathbf{F}\) is not eonservative. $$\mathbf{F}=-y^{2} \mathbf{i}+x^{2} \mathbf{j}$$

Compute \(\partial f / \partial x\) and \(\partial f / \partial y\) in \(11-18\). Draw the gradient field \(\mathbf{F}=\) grad \(f\) and the equipotentials \(f(x, y)=c:\) $$f=e^{x} \cos y$$

Explain why \(\iint\) curl \(\mathbf{F} \cdot \mathbf{n} d S=0\) over the closed boundary of any solid \(V\).

Compute \(\partial f / \partial x\) and \(\partial f / \partial y\) in \(11-18\). Draw the gradient field \(\mathbf{F}=\) grad \(f\) and the equipotentials \(f(x, y)=c:\) $$f=e^{x-y}$$

Suppose curl \(\mathbf{F}=0\) and div \(\mathbf{F}=0 .\) (a) Why is \(\mathbf{F}\) the gradient of a potential? (b) Why does the potential satisfy Laplace's equation \(f_{x x}+f_{y y}+f_{z z}=0^{2}\)

Compute the surface integrals \(\iint g(x, y, z) d S\). \(g=x\) on the cylinder \(x^{2}+y^{2}=4\) between \(z=0\) and \(z=3\).

Find a potential \(f\) if it exists. $$\mathbf{F}=z \mathbf{i}+\mathbf{j}+x \mathbf{k}$$

Describe the closed surface \(S\) and outward normal \(n\) : (a) \(V=\) hollow ball \(1 \leqslant x^{2}+y^{2}+z^{2} \leqslant 9\) (b) \(V=\) solid cylinder \(x^{2}+y^{2} \leqslant 1,|z| \leqslant 7\). (c) \(V=\) pyramid \(x \geqslant 0, y \geqslant 0, z \geqslant 0, x+2 y+3 z \leqslant 1\) (d) \(V=\) solid cone \(x^{2}+y^{2} \leqslant z^{2} \leqslant 1\).

Compute both sides of \(\oint \mathrm{F} \cdot \mathrm{a} d s=\iint\left(M_{x}+N_{y}\right) d x d y\) in 15-20. \(\mathbf{F}=x^{2} y \mathbf{j}\) in the unit triangle (sides \(\left.x=0, y=0, x+y=1\right)\).

\(\mathrm{~A}\) wire hoop around a vertical circle \(x^{2}+z^{2}=a^{2}\) has density \(\rho=a+z\). Find its mass \(M=\int \rho d s\).

Find a potential \(f\) if it exists. $$\mathbf{F}=2 x y z \mathbf{i}+x^{2} z \mathbf{j}+x^{2} y \mathbf{k}$$

A wire of constant density \(\rho\) lies on the semicircle \(x^{2}+y^{2}=a^{2}, y \geqslant 0 .\) Find its mass \(M\) and also its moment \(M_{x}=\int \rho y d s .\) Where is its center of mass \(\bar{x}=M_{y} / M, \bar{y}=M_{x} /\) \(M ?\)

If the density around the circle \(x^{2}+y^{2}=a^{2}\) is \(\rho=x^{2}\), what is the mass and where is the center of mass?

Find a potential \(f\) if it exists. $$\mathbf{F}=y z \mathbf{i}+x z \mathbf{j}+\left(x y+z^{2}\right) \mathbf{k}$$

Is it possible to have \(\mathbf{F} \cdot \mathbf{n}=0\) at all points of \(S\) and also \(\operatorname{div} \mathbf{F}=0\) at all points in \(\boldsymbol{V} ? \mathbf{F}=\mathbf{0}\) is not allowed.

Find \(\int \mathbf{F} \cdot d \mathbf{R}\) along the space curve \(x=t, y=t^{2}, z=t^{3}\) \(0 \leqslant t \leqslant 1\) (a) \(\mathbf{F}=\operatorname{grad}(x y+x z)\) (b) \(\mathbf{F}=y \mathbf{i}-x \mathbf{j}+z \mathbf{k}\)

Inside a solid ball (radius \(a\), density \(1,\) mass \(\left.M=4 \pi a^{3} / 3\right)\) the gravity field is \(\mathbf{F}=-G M \mathbf{R} / a^{3}\). (a) Check div \(\mathbf{F}=-4 \pi G\) in Gauss's Law. (b) The force at the surface is the same as if the whole mass \(M\) were _____ (c) Find a gradient field with \(\operatorname{div} \mathbf{F}=6\) in the balt \(\rho \leqslant a\) and \(\operatorname{div} \mathbf{F}=0\) outside.

(a) Find the unit tangent vector \(\mathrm{T}\) and the speed \(d s / d t\) along the path \(\mathbf{R}=2 t \mathbf{i}+t^{2} \mathbf{j}\) (b) For \(\mathbf{F}=3 x \mathbf{i}+4 \mathbf{j},\) find \(\mathbf{F} \cdot \mathbf{T} d s\) using (a) and \(\mathbf{F} \cdot d \mathbf{R}\) directly. (c) What is the work from (2,1) to (4,4)\(?\)

Inside a solid ball (radius \(a\), density \(1,\) mass \(\left.M=4 \pi a^{3} / 3\right)\) the gravity field is \(\mathbf{F}=-G M \mathbf{R} / a^{3}\). (a) Check div \(\mathbf{F}=-4 \pi G\) in Gauss's Law. (b) The force at the surface is the same as if the whole mass \(M\) were _____ (c) Find a gradient field with div \(\mathbf{F}=6\) in the balt \(\rho \leqslant a\) and \(\operatorname{div} \mathbf{F}=0\) outside.

The field \(\mathbf{R} / r^{2}\) in Example 7 has zero divergence except at \(r=0 .\) Solve \(\partial g / \partial y=x /\left(x^{2}+y^{2}\right)\) to find an attempted stream function \(g\). Does \(g\) have trouble at the origin?