Chapter 16

In Exercises \(1-16,\) 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.) $$ \text {The paraboloid } z=x^{2}+y^{2}, z \leq 4 $$

In Exercises \(1-8,\) find the divergence of the field. $$ \mathbf{F}=(x-y+z) \mathbf{i}+(2 x+y-z) \mathbf{j}+(3 x+2 y-2 z) \mathbf{k} $$

In Exercises \(1-6,\) find the curl of each vector field \(\mathbf{F}\) $$\mathbf{F}=(x+y-z) \mathbf{i}+(2 x-y+3 z) \mathbf{j}+(3 x+2 y+z) \mathbf{k}$$

In Exercises \(1-8,\) integrate the given function over the given surface. Parabolic cylinder \(G(x, y, z)=x,\) over the parabolic cylinder \(y=x^{2}, 0 \leq x \leq 2,0 \leq z \leq 3\)

Find the \(\mathbf{k}\) -component of \(\operatorname{curl}(\mathbf{F})\) for the following vector fields on the plane. \(\mathbf{F}=(x+y) \mathbf{i}+(2 x y) \mathbf{j}\)

Find the gradient fields of the functions in Exercises \(1-4\) $$f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2}$$

In Exercises \(1-16,\) 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.) $$ \text {The paraboloid }z=9-x^{2}-y^{2}, z \geq 0 $$

Find the \(\mathbf{k}\) -component of \(\operatorname{curl}(\mathbf{F})\) for the following vector fields on the plane. \(\mathbf{F}=\left(x^{2}-y\right) \mathbf{i}+\left(y^{2}\right) \mathbf{j}\)

In Exercises \(1-6,\) find the curl of each vector field \(\mathbf{F}\) $$\mathbf{F}=\left(x^{2}-y\right) \mathbf{i}+\left(y^{2}-z\right) \mathbf{j}+\left(z^{2}-x\right) \mathbf{k}$$

In Exercises \(1-8,\) find the divergence of the field. $$ \mathbf{F}=(x \ln y) \mathbf{i}+(y \ln z) \mathbf{j}+(z \ln x) \mathbf{k} $$

In Exercises \(1-8,\) integrate the given function over the given surface. Circular cylinder \(G(x, y, z)=z,\) over the cylindrical surface \(y^{2}+z^{2}=4, z \geq 0,1 \leq x \leq 4\)

Find the gradient fields of the functions in Exercises \(1-4\) $$f(x, y, z)=\ln \sqrt{x^{2}+y^{2}+z^{2}}$$

Find the \(\mathbf{k}\) -component of \(\operatorname{curl}(\mathbf{F})\) for the following vector fields on the plane. \(\mathbf{F}=\left(x e^{y}\right) \mathbf{i}+\left(y e^{x}\right) \mathbf{j}\)

In Exercises \(1-6,\) find the curl of each vector field \(\mathbf{F}\) $$\mathbf{F}=(x y+z) \mathbf{i}+(y z+x) \mathbf{j}+(x z+y) \mathbf{k}$$

In Exercises \(1-8,\) find the divergence of the field. $$ \mathbf{F}=y e^{x y z} \mathbf{i}+z e^{x y z} \mathbf{j}+x e^{x y z} \mathbf{k} $$

In Exercises \(1-8,\) integrate the given function over the given surface. Sphere \(\quad G(x, y, z)=x^{2},\) over the unit sphere \(x^{2}+y^{2}+z^{2}=1\)

Find the gradient fields of the functions in Exercises \(1-4\) $$g(x, y, z)=e^{z}-\ln \left(x^{2}+y^{2}\right)$$

In Exercises \(1-16,\) 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.) $$ \begin{array}{l}{\text { Cone frustum The first-octant portion of the cone } z=} \\ {\sqrt{x^{2}+y^{2}} / 2 \text { between the planes } z=0 \text { and } z=3}\end{array} $$

In Exercises \(1-16,\) 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.) $$ \begin{array}{l}{\text { Cone frustum The portion of the cone } z=2 \sqrt{x^{2}+y^{2}} \text { between }} \\ {\text { the planes } z=2 \text { and } z=4}\end{array} $$

Find the \(\mathbf{k}\) -component of \(\operatorname{curl}(\mathbf{F})\) for the following vector fields on the plane. \(\mathbf{F}=\left(x^{2} y\right) \mathbf{i}+\left(x y^{2}\right) \mathbf{j}\)

In Exercises \(1-6,\) find the curl of each vector field \(\mathbf{F}\) $$\mathbf{F}=y e^{2} \mathbf{i}+z e^{x} \mathbf{j}-x e^{y} \mathbf{k}$$

In Exercises \(1-8,\) find the divergence of the field. $$ \mathbf{F}=\sin (x y) \mathbf{i}+\cos (y z) \mathbf{j}+\tan (x z) \mathbf{k} $$

Find the gradient fields of the functions in Exercises \(1-4\) $$g(x, y, z)=x y+y z+x z$$

In Exercises \(1-16,\) 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.) $$ \begin{array}{l}{\text { Spherical cap The cap cut from the sphere } x^{2}+y^{2}+z^{2}=9} \\ {\text { by the cone } z=\sqrt{x^{2}+y^{2}}}\end{array} $$

Find the \(\mathbf{k}\) -component of \(\operatorname{curl}(\mathbf{F})\) for the following vector fields on the plane. \(\mathbf{F}=(y \sin x) \mathbf{i}+(x \sin y) \mathbf{j}\)

In Exercises \(1-6,\) find the curl of each vector field \(\mathbf{F}\) $$\mathbf{F}=x^{2} y z \mathbf{i}+x y^{2} z \mathbf{j}+x y z^{2} \mathbf{k}$$

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 \((0,0) . )\)

In Exercises \(1-8,\) integrate the given function over the given surface. Portion of plane \(F(x, y, z)=z,\) over the portion of the plane \(x+y+z=4\) that lies above the square \(0 \leq x \leq 1\) \(0 \leq y \leq 1,\) in the \(x y\) -plane

In Exercises \(1-16,\) 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.) $$ \begin{array}{l}{\text { Spherical cap The portion of the sphere } x^{2}+y^{2}+z^{2}=4 \text { in }} \\ {\text { the first octant between the } x y \text { -plane and the cone } z=\sqrt{x^{2}+y^{2}}}\end{array} $$

Find the \(\mathbf{k}\) -component of \(\operatorname{curl}(\mathbf{F})\) for the following vector fields on the plane. \(\mathbf{F}=(x / y) \mathbf{i}-(y / x) \mathbf{j}\)

In Exercises \(1-6,\) find the curl of each vector field \(\mathbf{F}\) $$\mathbf{F}=\frac{x}{y z} \mathbf{i}-\frac{y}{x z} \mathbf{j}+\frac{z}{x y} \mathbf{k}$$

In Exercises \(1-8,\) integrate the given function over the given surface. Cone \(\quad F(x, y, z)=z-x, \quad\) over \(\quad\) the cone \(\quad z=\sqrt{x^{2}+y^{2}}\) \(0 \leq z \leq 1\)

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 \(|\mathbf{F}|=\) \(\sqrt{a^{2}+b^{2}}\)

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 \(C : \mathbf{r}=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq 2 \pi.\) \(\mathbf{F}=-y \mathbf{i}+x \mathbf{j}\)

In Exercises \(1-16,\) 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.) $$ \begin{array}{l}{\text { Spherical band The portion of the sphere } x^{2}+y^{2}+z^{2}=3} \\ {\text { between the planes } z=\sqrt{3} / 2 \text { and } z=-\sqrt{3} / 2}\end{array} $$

In Exercises \(7-12,\) find a potential function \(f\) for the field \(\mathbf{F}\). $$\mathbf{F}=2 x \mathbf{i}+3 y \mathbf{j}+4 z \mathbf{k}$$

In Exercises \(1-8,\) integrate the given function over the given surface. Parabolic dome \(H(x, y, z)=x^{2} \sqrt{5-4 z},\) over the parabolic dome \(z=1-x^{2}-y^{2}, z \geq 0\)

In Exercises \(7-12\) , 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{equation} \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)=\mathbf{i} \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} \end{equation} $$\mathbf{F}=3 \mathrm{yi}+2 x \mathbf{j}+4 z \mathbf{k}$$

In Exercises \(7-12,\) 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{equation} \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} \end{equation}

Match the vector equations in Exercises \(1 - 8\) with the graphs \(( a ) - ( h )\) given here. $$ \mathbf { r } ( t ) = \left( t ^ { 2 } - 1 \right) \mathbf { j } + 2 t \mathbf { k } , \quad - 1 \leq t \leq 1 $$

In Exercises \(1-16,\) 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.) $$ \begin{array}{l}{\text { Spherical cap The upper portion cut from the sphere }} \\ {x^{2}+y^{2}+z^{2}=8 \text { by the plane } z=-2}\end{array} $$

In Exercises \(7-12,\) 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{equation} \begin{array}{l}{\mathbf{F}=2 y \mathbf{i}+3 x \mathbf{j}-z^{2} \mathbf{k}} \\\ {\text { C: The circle } x^{2}+y^{2}=9 \text { in the } x y \text { -plane, counterclockwise }} \\ {\text { when viewed from above }}\end{array} \end{equation}

In Exercises \(7-12,\) find a potential function \(f\) for the field \(\mathbf{F}\). $$\mathbf{F}=(y+z) \mathbf{i}+(x+z) \mathbf{j}+(x+y) \mathbf{k}$$

In Exercises \(1-8,\) integrate the given function over the given surface. Spherical cap \(\quad H(x, y, z)=y z,\) over the part of the sphere \(x^{2}+y^{2}+z^{2}=4\) that lies above the cone \(z=\sqrt{x^{2}+y^{2}}\)

Match the vector equations in Exercises \(1 - 8\) with the graphs \(( a ) - ( h )\) given here. $$ \mathbf { r } ( t ) = ( 2 \cos t ) \mathbf { i } + ( 2 \sin t ) \mathbf { k } , \quad 0 \leq t \leq \pi $$

In Exercises \(1-16,\) 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.) $$ \begin{array}{l}{\text { Parabolic cylinder between planes } \text { planes } \text { The surface cut from the }} \\ {\text { parabolic cylinder } z=4-y^{2} \text { by the planes } x=0, x=2, \text { and }} \\\ {z=0}\end{array} $$

In Exercises \(9-20,\) use the Divergence Theorem to find the outward flux of \(\mathbf{F}\) across the boundary of the region \(D .\) Cube \(\mathbf{F}=(y-x) \mathbf{i}+(z-y) \mathbf{j}+(y-x) \mathbf{k}\) \(D :\) The cube bounded by the planes \(x=\pm 1, y=\pm 1,\) and \(\quad z=\pm 1\)

In Exercises \(7-12,\) 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{equation} \begin{array}{l}{\mathbf{F}=y \mathbf{i}+x z \mathbf{j}+x^{2} \mathbf{k}} \\\ {\text { C: 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} \end{equation}

In Exercises \(7-12,\) find a potential function \(f\) for the field \(\mathbf{F}\). $$\mathbf{F}=e^{y+2 z}(\mathbf{i}+x \mathbf{j}+2 x \mathbf{k})$$

Evaluate \(\int _ { C } ( x + y ) d s\) where \(C\) is the straight-line segment \(x = t , y = ( 1 - t ) , z = 0 ,\) from \(( 0,1,0 )\) to \(( 1,0,0 )\)