Chapter 16

Integrate the given function over the given surface. \begin{equation}\begin{array}{l}{\text { Parabolic cylinder } G(x, y, z)=x, \text { over the parabolic cylinder }} \\ {y=x^{2}, 0 \leq x \leq 2,0 \leq z \leq 3}\end{array}\end{equation}

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) The paraboloid \(z=x^{2}+y^{2}, z \leq 4\)

In Exercises \(1-4,\) verify the conclusion of Green's Theorem by evaluating both sides of Equations \((3)\) and \((4)\) for the field \(\mathbf{F}=M \mathbf{i}+N \mathbf{j}\) . Take the domains of integration in each case to be the disk \(R : x^{2}+y^{2} \leq a^{2}\) and its bounding circle \(C : \mathbf{r}=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq 2 \pi\) $$\mathbf{F}=-y \mathbf{i}+x \mathbf{j}$$

Find the gradient fields of the functions $$ f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2} $$

Which fields are conservative, and which are not? \(\mathbf{F}=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k}\)

In Exercises \(1-6,\) use the surface integral in Stokes' Theorem to calculate the circulation of the field \(\mathbf{F}\) around the curve \(C\) in the indicated direction. \(\begin{array}{l}{\mathbf{F}=x^{2} \mathbf{i}+2 x \mathbf{j}+z^{2} \mathbf{k}} \\\ {C : \text { The ellipse } 4 x^{2}+y^{2}=4 \text { in the } x y \text { -plane, counterclockwise }} \\ {\text { when viewed from above }}\end{array}\)

In Exercises \(1-6,\) use the surface integral in Stokes' Theorem to calculate the circulation of the field \(\mathbf{F}\) around the curve \(C\) in the indicated direction. \(\begin{array}{l}{\mathbf{F}=2 \mathrm{yi}+3 x \mathbf{j}-z^{2} \mathbf{k}} \\\ {C : \text { The circle } x^{2}+y^{2}=9 \text { in the } x y \text { -plane, counterclockwise }} \\ {\text { when viewed from above }}\end{array}\)

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) The paraboloid \(z=9-x^{2}-y^{2}, z \geq 0\)

Find the gradient fields of the functions $$ f(x, y, z)=\ln \sqrt{x^{2}+y^{2}+z^{2}} $$

Which fields are conservative, and which are not? \(\mathbf{F}=(y \sin z) \mathbf{i}+(x \sin z) \mathbf{j}+(x y \cos z) \mathbf{k}\)

In Exercises \(1-4,\) verify the conclusion of Green's Theorem by evaluating both sides of Equations \((3)\) and \((4)\) for the field \(\mathbf{F}=M \mathbf{i}+N \mathbf{j}\) . Take the domains of integration in each case to be the disk \(R : x^{2}+y^{2} \leq a^{2}\) and its bounding circle \(C : \mathbf{r}=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq 2 \pi\) $$\mathbf{F}=y \mathbf{i}$$

Integrate the given function over the given surface. \begin{equation}\begin{array}{l} {\text { Sphere } \quad G(x, y, z)=x^{2}, \text { over the unit sphere } x^{2}+y^{2}+z^{2}=1}\end{array}\end{equation}

In Exercises \(1-6,\) use the surface integral in Stokes' Theorem to calculate the circulation of the field \(\mathbf{F}\) around the curve \(C\) in the indicated direction. \(\begin{array}{l}{\mathbf{F}=y \mathbf{i}+x z \mathbf{j}+x^{2} \mathbf{k}} \\\ {C : \text { The boundary of the triangle cut from the plane } x+y+z=1} \\\ {\text { by the first octant, counterclockwise when viewed from above }}\end{array}\)

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Cone frustum The first-octant portion of the cone \(z=\) \(\sqrt{x^{2}+y^{2}} / 2\) between the planes \(z=0\) and \(z=3\)

Find the gradient fields of the functions $$ g(x, y, z)=e^{z}-\ln \left(x^{2}+y^{2}\right) $$

Which fields are conservative, and which are not? \(\mathbf{F}=y \mathbf{i}+(x+z) \mathbf{j}-y \mathbf{k}\)

In Exercises \(1-4,\) verify the conclusion of Green's Theorem by evaluating both sides of Equations \((3)\) and \((4)\) for the field \(\mathbf{F}=M \mathbf{i}+N \mathbf{j}\) . Take the domains of integration in each case to be the disk \(R : x^{2}+y^{2} \leq a^{2}\) and its bounding circle \(C : \mathbf{r}=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq 2 \pi\) $$\mathbf{F}=2 x \mathbf{i}-3 y \mathbf{j}$$

Integrate the given function over the given surface. \begin{equation}\begin{array}{l}{\text { Hemisphere } G(x, y, z)=z^{2}, \text { over the hemisphere } x^{2}+y^{2}+} \\ {z^{2}=a^{2}, z \geq 0}\end{array}\end{equation}

In Exercises \(1-6,\) use the surface integral in Stokes' Theorem to calculate the circulation of the field \(\mathbf{F}\) around the curve \(C\) in the indicated direction. \(\begin{array}{l}{\mathbf{F}=\left(y^{2}+z^{2}\right) \mathbf{i}+\left(x^{2}+z^{2}\right) \mathbf{j}+\left(x^{2}+y^{2}\right) \mathbf{k}} \\ {C : \text { The boundary of the triangle cut from the plane } x+y+z=1} \\ {\text { by the first octant, counterclockwise when viewed from above }}\end{array}\)

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Spherical cap The cap cut from the sphere \(x^{2}+y^{2}+z^{2}=9\) by the cone \(z=\sqrt{x^{2}+y^{2}}\)

In Exercises \(1-4,\) verify the conclusion of Green's Theorem by evaluating both sides of Equations \((3)\) and \((4)\) for the field \(\mathbf{F}=M \mathbf{i}+N \mathbf{j}\) . Take the domains of integration in each case to be the disk \(R : x^{2}+y^{2} \leq a^{2}\) and its bounding circle \(C : \mathbf{r}=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq 2 \pi\) $$\mathbf{F}=-x^{2} y \mathbf{i}+x y^{2} \mathbf{j}$$

Find the gradient fields of the functions $$ g(x, y, z)=x y+y z+x z $$

Which fields are conservative, and which are not? \(\mathbf{F}=-y \mathbf{i}+x \mathbf{j}\)

Integrate the given function over the given surface. \begin{equation}\begin{array}{l}{\text { Portion of plane } F(x, y, z)=z, \text { over the portion of the plane }} \\ {x+y+z=4 \text { that lies above the square } 0 \leq x \leq 1,} \\ {0 \leq y \leq 1, \text { in the } x y \text { -plane }}\end{array}\end{equation}

In Exercises \(1-6,\) use the surface integral in Stokes' Theorem to calculate the circulation of the field \(\mathbf{F}\) around the curve \(C\) in the indicated direction. \(\begin{array}{l}{\mathbf{F}=\left(y^{2}+z^{2}\right) \mathbf{i}+\left(x^{2}+y^{2}\right) \mathbf{j}+\left(x^{2}+y^{2}\right) \mathbf{k}} \\ {C : \text { The square bounded by the lines } x=\pm 1 \text { and } y=\pm 1 \text { in the }} \\ {x y \text { -plane, counterclockwise when viewed from above }}\end{array}\)

In Exercises \(5-16\) , use the Divergence Theorem to find the outward flux of \(\mathbf{F}\) across the boundary of the region \(D .\) Cube \(\quad \mathbf{F}=(y-x) \mathbf{i}+(z-y) \mathbf{j}+(y-x) \mathbf{k}\) \(D :\) The cube bounded by the planes \(x=\pm 1, y=\pm 1,\) and \(z=\pm 1\)

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Spherical cap The cap cut from the sphere \(x^{2}+y^{2}+z^{2}=9\) by the cone \(z=\sqrt{x^{2}+y^{2}}\)

In Exercises \(5-14,\) use Green's Theorem to find the counterclockwise circulation and outward flux for the field \(F\) and curve \(C .\) $$ \begin{array}{l}{\mathbf{F}=(x-y \mathbf{i}+(y-x) \mathbf{j}} \\ {\text { C: The square bounded by } x=0, x=1, y=0, y=1}\end{array} $$

Give a formula \(\mathbf{F}=M(x, y) \mathbf{i}+N(x, y) \mathbf{j}\) for the vector field in the plane that has the property that \(\mathbf{F}\) points toward the origin with magnitude inversely proportional to the square of the distance from \((x, y)\) to the origin. (The field is not defined at \((0,0) . )\)

Which fields are conservative, and which are not? \(\mathbf{F}=(z+y) \mathbf{i}+z \mathbf{j}+(y+x) \mathbf{k}\)

Integrate the given function over the given surface. \begin{equation}\begin{array}{l}{\text { Cone } \quad F(x, y, z)=z-x, \quad \text { over } \quad \text { the cone } \quad z=\sqrt{x^{2}+y^{2}}}, \\ {0 \leq z \leq 1}\end{array}\end{equation}

In Exercises \(1-6,\) use the surface integral in Stokes' Theorem to calculate the circulation of the field \(\mathbf{F}\) around the curve \(C\) in the indicated direction. \(\begin{array}{l}{\mathbf{F}=x^{2} y^{3} \mathbf{i}+\mathbf{j}+z \mathbf{k}} \\\ {C : \text { The intersection of the cylinder } x^{2}+y^{2}=4 \text { and the hemisphere }} \\ {x^{2}+y^{2}+z^{2}=16, z \geq 0, \text { counterclockwise when viewed from }} \\ {\text { above }}\end{array}\)

In Exercises \(5-16\) , use the Divergence Theorem to find the outward flux of \(\mathbf{F}\) across the boundary of the region \(D .\) $$\mathbf{F}=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k}$$ a. Cube \(D :\) The cube cut from the first octant by the planes \(x=1, y=1,\) and \(z=1\) b. Cube \(D :\) The cube bounded by the planes \(x=\pm 1\) \(y=\pm 1,\) and \(z=\pm 1\) c. Cylindrical can \(D :\) The region cut from the solid cylinder \(x^{2}+y^{2} \leq 4\) by the planes \(z=0\) and \(z=1\)

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Spherical cap The portion of the sphere \(x^{2}+y^{2}+z^{2}=4\) in the first octant between the \(x y\) -plane and the cone \(z=\sqrt{x^{2}+y^{2}}\)

In Exercises \(5-14,\) use Green's Theorem to find the counterclockwise circulation and outward flux for the field \(F\) and curve \(C .\) $$ \begin{array}{l}{\mathbf{F}=\left(x^{2}+4 y\right) \mathbf{i}+\left(x+y^{2}\right) \mathbf{j}} \\ {C : \text { The square bounded by } x=0, x=1, y=0, y=1}\end{array} $$

Give a formula \(\mathbf{F}=M(x, y) \mathbf{i}+N(x, y) \mathbf{j}\) for the vector field in the plane that has the properties that \(\mathbf{F}=0\) at \((0,0)\) and that at any other point \((a, b), \mathbf{F}\) is tangent to the circle \(x^{2}+y^{2}=a^{2}+b^{2}\) and points in the clockwise direction with magnitude \(|\mathbf{F}|=\) \(\sqrt{a^{2}+b^{2}}\)

Which fields are conservative, and which are not? \(\mathbf{F}=\left(e^{x} \cos y\right) \mathbf{i}-\left(e^{x} \sin y\right) \mathbf{j}+z \mathbf{k}\)

In Exercises \(5-16\) , use the Divergence Theorem to find the outward flux of \(\mathbf{F}\) across the boundary of the region \(D .\) Cylinder and paraboloid \(\mathbf{F}=y \mathbf{i}+x y \mathbf{j}-z \mathbf{k}\) \(D :\) The region inside the solid cylinder \(x^{2}+y^{2} \leq 4\) between the plane \(z=0\) and the paraboloid \(z=x^{2}+y^{2}\)

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Spherical band The portion of the sphere \(x^{2}+y^{2}+z^{2}=3\) between the planes \(z=\sqrt{3} / 2\) and \(z=-\sqrt{3} / 2\)

In Exercises \(5-14,\) use Green's Theorem to find the counterclockwise circulation and outward flux for the field \(F\) and curve \(C .\) $$ \begin{array}{l}{\mathbf{F}=\left(y^{2}-x^{2}\right) \mathbf{i}+\left(x^{2}+y^{2}\right) \mathbf{j}} \\ {C : \text { The triangle bounded by } y=0, x=3, \text { and } y=x}\end{array} $$

Find a potential function \(f\) for the field \(\mathbf{F}.\) \(\mathbf{F}=2 x \mathbf{i}+3 y \mathbf{j}+4 z \mathbf{k}\)

Integrate the given function over the given surface. \begin{equation}\begin{array}{l}{\text { Parabolic dome } H(x, y, z)=x^{2} \sqrt{5-4 z}, \text { over the parabolic }} \\ {\text { dome } z=1-x^{2}-y^{2}, z \geq 0}\end{array}\end{equation}

Find the line integrals of \(\mathbf{F}\) from \((0,0,0)\) to \((1,1,1)\) over each of the following paths in the accompanying figure. $$ \begin{array}{l}{\text { a. The straight-line path } C_{1} : \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1} \\ {\text { b. The curved path } C_{2} : \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{4} \mathbf{k}, \quad 0 \leq t \leq 1} \\ {\text { c. The path } C_{3} \cup C_{4} \text { consisting of the line segment from }(0,0,0)} \\ {\text { to }(1,1,0) \text { followed by the segment from }(1,1,0) \text { to }(1,1,1)}\end{array} $$ $$ \mathbf{F}=3 y \mathbf{i}+2 x \mathbf{j}+4 z \mathbf{k} $$

Integrate the given function over the given surface. \begin{equation}\begin{array}{l}{\text { Spherical cap } H(x, y, z)=y z, \text { over the part of the sphere }} \\ {x^{2}+y^{2}+z^{2}=4 \text { that lies above the cone } z=\sqrt{x^{2}+y^{2}}}\end{array}\end{equation}

Let \(\mathbf{n}\) be the outer unit normal (normal away from the origin) of the parabolic shell $$S : 4 x^{2}+y+z^{2}=4, \quad y \geq 0$$ and let $$\mathbf{F}=\left(-z+\frac{1}{2+x}\right) \mathbf{i}+\left(\tan ^{-1} y\right) \mathbf{j}+\left(x+\frac{1}{4+z}\right) \mathbf{k}.$$ Find the value of $$\iint_{S} \nabla \times \mathbf{F} \cdot \mathbf{n} d \sigma.$$

Along the curve \(\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}-(\cos t) \mathbf{k}, 0 \leq t \leq \pi\) evaluate each of the following integrals. $$ \text { a. } \int_{c} x z d x \quad \text { b. } \int_{c} x z d y \quad \text { c. } \int_{c} x y z d z $$

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Spherical cap The upper portion cut from the sphere \(x^{2}+y^{2}+z^{2}=8\) by the plane \(z=-2\)

In Exercises \(5-14,\) use Green's Theorem to find the counterclockwise circulation and outward flux for the field \(F\) and curve \(C .\) $$ \begin{array}{l}{\mathbf{F}=(x+y) \mathbf{i}-\left(x^{2}+y^{2}\right) \mathbf{j}} \\ {C : \text { The triangle bounded by } y=0, x=1, \text { and } y=x}\end{array} $$

Find a potential function \(f\) for the field \(\mathbf{F}.\) \(\mathbf{F}=(y+z) \mathbf{i}+(x+z) \mathbf{j}+(x+y) \mathbf{k}\)

Integrate \(G(x, y, z)=x+y+z\) over the surface of the cube cut from the first octant by the planes \(x=a, y=a, z=a\).