Chapter 14

Find the gradient field corresponding to \(f\) Use a CAS to graph it. $$f(x, y)=x^{2}+y^{2}$$

Use the Divergence Theorem to compute \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S\). \(Q \quad\) is bounded by \(\quad x^{2}+y^{2}=1, z=0 \quad\) and \(\quad z=1\) \(\mathbf{F}=\left\langle x-y^{3}, x^{2} \sin z, 3 z\right\rangle\)

Let 1 be the current crossing an open surface \(S\), so that \(I=\iint_{S} \mathbf{J} \cdot \mathbf{n} d S .\) Given that \(I=f_{C} \mathbf{B} \cdot d \mathbf{r}\) (where \(C\) is the positively oriented boundary of \(S\) ), show that \(\mathbf{J}=\nabla \times \mathbf{B}\).

Sketch a graph of the parametric surface. \(x=2 \sin u \cos v, y=2 \sin u \sin v, z=2 \cos u\)

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(\mathbf{F}=\left\langle e^{x^{2}}-y, e^{2 x}+y\right\rangle\) and \(C\) is formed by \(y=1-x^{2}\) and \(y=0\)

Show that the line integral is independent of path and use a potential function to evaluate the integral. $$\int_{C} 2 x y d x+\left(x^{2}-1\right) d y, \text { where \(C\) runs from \((1,0)\) to \((3,1)\) } $$

Evaluate the line integral. \(\int_{C} 3 y d s,\) where \(C\) is the portion of \(y=x^{2}\) from (0,0) to (2,4)

Use Stokes' Theorem to compute $$\begin{aligned}&\iint(\nabla \times \mathbf{F}) \cdot \mathbf{n} d \mathbf{S}\\\&S \end{aligned}$$ \(S\) is the portion of the cone \(z=\sqrt{x^{2}+y^{2}}\) inside the cylinder \(x^{2}+y^{2}=2,\) \(\mathbf{n}\) downward, \(\mathbf{F}=\left\langle z x, x^{2}+y^{2}, z^{2}-y^{2}\right\rangle\)

Determine whether the given vector field is conservative and/or incompressible. $$\left(2 x y, x^{2}-3 y^{2} z^{2}, 1-2 z y^{3}\right)$$

Find the gradient field corresponding to \(f\) Use a CAS to graph it. $$f(x, y)=x^{2}-y^{2}$$

Use the Divergence Theorem to compute \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S\). \(Q\) is bounded by \(x^{2}+y^{2}=4, z=1\) and \(z=8-y\) \(\mathbf{F}=\left\langle y^{2} z, 2 y-e^{z}, \sin x\right\rangle\)

Sketch a graph of the parametric surface. \(x=2 \cos v, y=2 \sin v, z=u\)

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(\mathbf{F}=\left\langle x e^{x y}+y, y e^{x y}+2 x\right\rangle\) and \(C\) is formed by \(y=x^{2}\) and \(y=4\)

Show that the line integral is independent of path and use a potential function to evaluate the integral. $$\int_{C} 3 x^{2} y^{2} d x+\left(2 x^{3} y-4\right) d y, \text { where \(C\) runs from \((1,2)\) to \((-1,1)\) }$$

Evaluate the line integral. \(\int_{C} 3 y d s,\) where \(C\) is the portion of \(y=x^{2}\) from (0,0) to (2,4)

Use Stokes' Theorem to evaluate \(\int c \mathbf{F} \cdot d \mathbf{r}\). \(C\) is the boundary of the portion of the paraboloid $$y=4-x^{2}-z^{2}$$ with $$y>0, \mathbf{n}$$ to the right, $$\mathbf{F}=\left\langle x^{2} z, 3 \cos y, 4 z^{3}\right\rangle$$

Determine whether the given vector field is conservative and/or incompressible. $$\left\langle 3 y z, x^{2}, x \cos y\right\rangle$$

Find the gradient field corresponding to \(f\) Use a CAS to graph it. $$f(x, y)=\sqrt{x^{2}+y^{2}}$$

Use the Divergence Theorem to compute \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S\). $$\begin{aligned} &Q \quad \text { is bounded by } \quad z=\sqrt{1-x^{2}-y^{2}} \quad \text { and } \quad z=0\\\ &\mathbf{F}=\left\langle x^{3}, y^{3}, z^{3}\right\rangle \end{aligned}$$

Use the electrostatic force \(\mathbf{E}=\frac{q}{4 \pi \epsilon_{0} r^{3}} \mathbf{r}\) for a charge \(q\) at the origin, where \(r=\langle x, y, z\rangle\) and \(r=\sqrt{x^{2}+y^{2}+z^{2}}\). If \(S\) is a closed surface not enclosing the origin, show that \(\iint_{S} \mathbf{E} \cdot \mathbf{n} d S=0\).

Sketch a graph of the parametric surface. \(x=u, y=\sin u \cos v, z=\sin u \sin v\)

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C}\left[y^{3}-\ln (x+1)\right] d x+(\sqrt{y^{2}+1}+3 x) d y,\) where \(C\) is formed by \(x=y^{2}\) and \(x=4\)

Show that the line integral is independent of path and use a potential function to evaluate the integral. $$\int_{C} y e^{x y} d x+\left(x e^{x y}-2 y\right) d y, \text { where \(C\) runs from \((1,0)\) to \((0,4)\) }$$

Evaluate the line integral. \(\int_{C} 2 x d x,\) where \(C\) is the portion of \(y=x^{2}\) from (2,4) to (0,0)

Use Stokes' Theorem to evaluate \(\int c \mathbf{F} \cdot d \mathbf{r}\). \(C\) is the boundary of the portion of the paraboloid \(x=y^{2}+z^{2}\) with \(x \leq 4,\) \(\mathbf{n}\) to the back, \(\mathbf{F}=\langle y z, y-4,2 x y\rangle\)

Determine whether the given vector field is conservative and/or incompressible. $$\left(y^{2}, x^{2} e^{z}, \cos x y\right)$$

Find the gradient field corresponding to \(f\) Use a CAS to graph it. $$f(x, y)=\sin \left(x^{2}+y^{2}\right)$$

Use the Divergence Theorem to compute \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S\). $$\begin{aligned} &\begin{array}{ccccccc} Q & \text { is } & \text { bounded } & \text { by } & z=-\sqrt{4-x^{2}-y^{2}} & \text { and } & z=0 \end{array}\\\ &\mathbf{F}=\left\langle x^{3}, y^{3}, z^{3}\right\rangle \end{aligned}$$

Use the electrostatic force \(\mathbf{E}=\frac{q}{4 \pi \epsilon_{0} r^{3}} \mathbf{r}\) for a charge \(q\) at the origin, where \(r=\langle x, y, z\rangle\) and \(r=\sqrt{x^{2}+y^{2}+z^{2}}\). If \(S\) is the sphere \(x^{2}+y^{2}+z^{2}=1,\) show that \(\iint_{S} \mathbf{E} \cdot \mathbf{n} d S=\frac{q}{\epsilon_{0}}\).

Sketch a graph of the parametric surface. \(x=\cos u \cos v, y=u, z=\cos u \sin v\)

Show that the line integral is independent of path and use a potential function to evaluate the integral. $$\int_{C}\left(2 x e^{x^{2}}-2 y\right) d x+(2 y-2 x) d y, \text { where \(C\) runs from \((1,2)\) to \((-1,1)\) }$$

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C}\left(y \sec ^{2} x-2\right) d x+\left(\tan x-4 y^{2}\right) d y,\) where \(C\) is formed by \(x=1-y^{2}\) and \(x=0\)

Evaluate the line integral. \(\int_{C} 3 y^{2} d y,\) where \(C\) is the portion of \(y=x^{2}\) from (2,4) to (0,0)

Use Stokes' Theorem to evaluate \(\int c \mathbf{F} \cdot d \mathbf{r}\). \(C\) is the boundary of the portion of \(z=4-x^{2}-y^{2}\) above the \(x y\) -plane, oriented upward, \(\mathbf{F}=\left\langle x^{2} e^{x}-y, \sqrt{y^{2}+1}, z^{3}\right\rangle\)

Determine whether the given vector field is conservative and/or incompressible. $$\left(\sin z, z^{2} e^{y z^{2}}, x \cos z+2 y z e^{y z^{2}}\right)$$

Find the flux of \(\mathbf{F}\) over \(\partial Q\) . $$\begin{aligned} &Q \text { is bounded by } z=\sqrt{x^{2}+y^{2}} \text { and } z=\sqrt{2-x^{2}-y^{2}}\\\ &\mathbf{F}=\left\langle x^{2}, z^{2}-x, y^{3}\right\rangle \end{aligned}$$

Find the gradient field corresponding to \(f\) Use a CAS to graph it. $$f(x, y)=x e^{-y}$$

Use the electrostatic force \(\mathbf{E}=\frac{q}{4 \pi \epsilon_{0} r^{3}} \mathbf{r}\) for a charge \(q\) at the origin, where \(r=\langle x, y, z\rangle\) and \(r=\sqrt{x^{2}+y^{2}+z^{2}}\). If \(S\) is the sphere \(x^{2}+y^{2}+z^{2}=R^{2},\) show directly that \(\iint_{S} \mathbf{E} \cdot \mathbf{n} d S=\frac{q}{\epsilon_{0}}\).

Show that the line integral is independent of path and use a potential function to evaluate the integral. $$\int_{C}\left(z^{2}+2 x y\right) d x+x^{2} d y+2 x z d z, \text { where \(C\) runs from \((2,1,3)\) to \((4,-1,0)\) }$$

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C} x^{2} d x+2 x d y+(z-2) d z,\) where \(C\) is the triangle from (0,0,2) to (2,0,2) to (2,2,2) to (0,0,2)

Evaluate the line integral. \(\int_{c} 3 y d x,\) where \(C\) is the portion of \(x=y^{2}\) from (1,1) to (4,2)

Use Stokes' Theorem to evaluate \(\int c \mathbf{F} \cdot d \mathbf{r}\). \(C\) is the boundary of the portion of \(z=x^{2}+y^{2}\) below \(z=4\) oriented downward, \(\mathbf{F}=\left\langle x^{2}, y^{4}-x, z^{2} \sin z\right\rangle\)

Determine whether the given vector field is conservative and/or incompressible. $$\left(2 x y \cos z, x^{2} \cos z-3 y^{2} z,-x^{2} y \sin z-y^{3}\right)$$

Find the flux of \(\mathbf{F}\) over \(\partial Q\). $$\begin{aligned} &Q \text { is bounded by } z=\sqrt{x^{2}+y^{2}} \text { and } z=\sqrt{8-x^{2}-y^{2}}\\\ &\mathbf{F}=\left\langle 3 x z^{2}, y^{3}, 3 z x^{2}\right\rangle \end{aligned}$$

Find the gradient field corresponding to \(f\) Use a CAS to graph it. $$f(x, y)=y \sin x$$

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C} 4 y d x+y^{3} d y+z^{4} d z,\) where \(C\) is \(x^{2}+y^{2}=4\) in the plane \(z=0\)

Show that the line integral is independent of path and use a potential function to evaluate the integral. $$\int_{C}\left(2 x \cos z-x^{2}\right) d x+(z-2 y) d y+\left(y-x^{2} \sin z\right) d z \text { where \(C\) runs from \((3,-2,0)\) to \((1,0, \pi)\) }$$

Evaluate the line integral. \(\int_{C}(x+y) d y,\) where \(C\) is the portion of \(x=y^{2}\) from (1,1) to (1,-1)

Use Stokes' Theorem to evaluate \(\int c \mathbf{F} \cdot d \mathbf{r}\). \(C\) is the intersection of \(z=x^{2}+y^{2}\) and \(z=8-y,\) oriented clockwise from above, \(\mathbf{F}=\left\langle 2 x^{2}, 4 y^{2}, e^{8 z^{2}}\right\rangle\)

Determine whether the given vector field is conservative and/or incompressible. $$\left(z^{2}-3 y e^{3 x}, z^{2}-e^{3 x}, 2 z \sqrt{x y}\right)$$