Chapter 14

Find the curl and divergence of the given vector field. $$\left\langle x^{2}, y-z, x e^{y}\right\rangle$$

Sketch several vectors in the vector field by hand and verify your sketch with a CAS. $$\mathbf{F}(x, y, z)=\langle 0, z, 1\rangle$$

Use the Divergence Theorem to compute \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S\). \begin{aligned} &Q \text { is the cube }-1 \leq x \leq 1,-1 \leq y \leq 1,-1 \leq z \leq 1\\\ &\mathbf{F}=\left\langle 4 y^{2}, 3 z-\cos x, z^{3}-x\right\rangle \end{aligned}

Use Gauss' Law for electricity and the relationship \(q=\iiint \int_{Q} \rho d V\). For \(\mathbf{E}=\left\langle 2 x y, y^{2}, 5 x y\right\rangle,\) find the total charge in the cone \(y=\sqrt{x^{2}+z^{2}}\).

Find a parametric representation of the surface. The portion of \(z=4-x^{2}-y^{2}\) above the \(x y\) -plane

Use Green's Theorem to evaluate the indicated line integral.. \(\oint_{C}\left(\frac{x}{x^{2}+1}-y\right) d x+(3 x-4 \tan y / 2) d y,\) where \(C\) is the portion of \(y=x^{2}\) from (-1,1) to \((1,1),\) followed by the portion of \(y=2-x^{2}\) from (1,1) to (-1,1)

Determine whether \(F\) is conservative. If it is, find a potential function \(f.\) $$\mathbf{F}(x, y)=\left\langle y e^{x y}, x e^{x y}+\cos y\right\rangle$$

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

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 \(z=\sqrt{4-x^{2}-y^{2}}\) above the \(x y\) -plane with \(\mathbf{n}\) upward, \(\mathbf{F}=\left\langle z x^{2}, z e^{x y^{2}}-x, x \ln y^{2}\right\rangle\)

Find the curl and divergence of the given vector field. $$\left(y, x^{2} y, 3 z+y\right)$$

Sketch several vectors in the vector field by hand and verify your sketch with a CAS. $$\mathbf{F}(x, y, z)=\langle 2,0,0\rangle$$

Use the Divergence Theorem to compute \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S\). \(Q\) is the rectangular box \(0 \leq x \leq 2,1 \leq y \leq 2,-1 \leq z \leq 2\) \(\mathbf{F}=\left\langle y^{3}-2 x, e^{x z}, 4 z\right\rangle\)

Use Gauss' Law for electricity and the relationship \(q=\iiint \int_{Q} \rho d V\). For \(\mathbf{E}=\langle 4 x-y, 2 y+z, 3 x y\rangle,\) find the total charge in the solid bounded by \(z=R-x^{2}-y^{2}\) and \(z=0\).

Find a parametric representation of the surface. The portion of \(z=x^{2}+y^{2}\) below \(z=4\)

Use Green's Theorem to evaluate the indicated line integral. \(\int_{C}\left(x y-e^{2 x}\right) d x+\left(2 x^{2}-4 y^{2}\right) d y,\) where \(C\) is formed by \(y=x^{2}\) and \(y=8-x^{2}\) oriented clockwise

Determine whether \(F\) is conservative. If it is, find a potential function \(f.\) $$\mathbf{F}(x, y)=\left\langle y \cos x y-2 x y, x \cos x y-x^{2}\right\rangle$$

Evaluate the line integral. \(\int_{C}(3 x-y) d s,\) where \(C\) is the quarter-circle \(x^{2}+y^{2}=9\) from (0,3) to (3,0)

Find the curl and divergence of the given vector field. $$\left\langle 3 y z, x^{2}, x \cos y\right\rangle$$

Sketch several vectors in the vector field by hand and verify your sketch with a CAS. $$\mathbf{F}(x, y, z)=\frac{\langle x, y, z\rangle}{\sqrt{x^{2}+y^{2}+z^{2}}}$$

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

Faraday showed that \(\quad \oint_{C} \mathbf{E} \cdot d \mathbf{r}=-\frac{d \phi}{d t}, \quad\) where \(\phi=\iint_{S} \mathbf{B} \cdot \mathbf{n} d S,\) for any capping surface \(S\) (that is, any positively oriented open surface with boundary \(C\) ). Use this to show that \(\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} .\) What mathematical assumption must be made?

Sketch a graph of the parametric surface. \(x=u, y=v, z=u^{2}+2 v^{2}\)

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C}\left(\tan x-y^{3}\right) d x+\left(x^{3}-\sin y\right) d y,\) where \(C\) is the circle \(x^{2}+y^{2}=2\)

Determine whether \(F\) is conservative. If it is, find a potential function \(f.\) $$\mathbf{F}(x, y, z)=\left\langle z^{2}+2 x y, x^{2}+1,2 x z-3\right\rangle$$

Evaluate the line integral. \(\int_{c} 2 x d x,\) where \(C\) is the quarter-circle \(x^{2}+y^{2}=4\) from (2,0) to (0,2)

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 \(y=x^{2}+z^{2}\) with \(y \leq 2,\) \(\mathbf{n}\) to the left, \(\mathbf{F}=\left\langle x y, 4 x e^{z^{2}}, y z+1\right\rangle\)

Find the curl and divergence of the given vector field. $$\left\langle y^{2}, x^{2} e^{z}, \cos x y\right\rangle$$

Sketch several vectors in the vector field by hand and verify your sketch with a CAS. $$\mathbf{F}(x, y, z)=\frac{\langle x, y, z)}{x^{2}+y^{2}+z^{2}}$$

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

If an electric field \(\mathbf{E}\) is conservative with potential function \(-\phi,\) use Gauss' Law of electricity to show that Poisson's equation must hold: \(\nabla^{2} \phi=-\frac{\rho}{\epsilon_{0}}\).

Sketch a graph of the parametric surface. \(x=u, y=v, z=4-u^{2}-v^{2}\)

Use Green's Theorem to evaluate the indicated line integral. \(\int_{C}\left(\sqrt{x^{2}+1}-x^{2} y\right) d x+\left(x y^{2}-y^{5 / 3}\right) d y,\) where \(C\) is the circle \(x^{2}+y^{2}=4\) oriented clockwise

Determine whether \(F\) is conservative. If it is, find a potential function \(f.\) $$\mathbf{F}(x, y, z)=\left\langle y^{2}-x, 2 x y+\sin z, y \cos z\right\rangle$$

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 unit cube \(0 \leq x \leq 1,0 \leq y \leq 1\) \(0 \leq z \leq 1 \quad\) with \(\quad z<1, \quad \mathbf{n} \quad\) upward \(\mathbf{F}=\left\langle x y z, 4 x^{2} y^{3}-z, 8 \cos x z^{2}\right\rangle\)

Find the curl and divergence of the given vector field. $$\left(2 x z, y+z^{2}, z y^{2}\right)$$

Use Maxwell's equation and \(\mathbf{J}=\rho \mathbf{v}\) to derive the continuity equation. (Hint: Start by computing \(\nabla \cdot \mathbf{J}\).) What mathematical assumption must be made?

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

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(\mathbf{F}=\left\langle x^{3}-y, x+y^{3}\right\rangle\) and \(C\) is formed by \(y=x^{2}\) and \(y=x\)

Determine whether \(F\) is conservative. If it is, find a potential function \(f.\) $$\mathbf{F}(x, y, z)=\left(y^{2} z^{2}+x, y+2 x y z^{2}, 2 x y^{2} z\right)$$

Evaluate the line integral. \(\int_{C} 3 y d x,\) where \(C\) is the half-ellipse \(x^{2}+4 y^{2}=4\) from (0,1) to (0,-1) with \(x \geq 0\)

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 unit cube \(0 \leq x \leq 1,0 \leq y \leq 1\) \(0 \leq z \leq 1\) with \(\quad z<1, \quad \mathbf{n} \quad\) downward, \(\mathbf{F}=\left\langle x y z, 4 x^{2} y^{3}-z, 8 \cos x z^{2}\right\rangle\)

Find the curl and divergence of the given vector field. $$\left(x y^{2}, 3 y^{2} z^{2}, 2 x-z y^{3}\right)$$

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

For a magnetic field \(\mathbf{B}\), Maxwell's equation \(\nabla \cdot \mathbf{B}=0\) implies that \(\mathbf{B}=\nabla \times \mathbf{A}\) for some vector field \(\mathbf{A} .\) Show that the flux of \(\mathbf{B}\) across an open surface \(S\) equals the circulation of \(\mathbf{A}\) around the closed curve \(C\), where \(C\) is the positively oriented boundary of \(S\).

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

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(\mathbf{F}=\left\langle y^{2}+3 x^{2} y, x y+x^{3}\right\rangle\) and \(C\) is formed by \(y=x^{2}\) and \(y=2 x\)

Determine whether \(F\) is conservative. If it is, find a potential function \(f.\) $$\mathbf{F}(x, y, z)=\left\langle 2 x e^{y z}-1, x^{2}+e^{y z}, x^{2} y e^{y z}\right\rangle$$

Evaluate the line integral. \(\int_{C} x^{2} d y,\) where \(C\) is the ellipse \(4 x^{2}+y^{2}=4\) oriented counterclockwise

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 \(z=\sqrt{x^{2}+y^{2}} \quad\) below the sphere \(x^{2}+y^{2}+z^{2}=2\) ,\(\mathbf{n}\) downward, $$\mathbf{F}=\left\langle x^{2}+y^{2}, z e^{x^{2}+y^{2}}, e^{x^{2}+z^{2}}\right\rangle$$

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