91Ó°ÊÓ

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 \(S\).

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

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 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, \(\mathbf{F}=\left\langle x y z, 4 x^{2} y^{3}-z, 8 \cos x z^{2}\right\rangle\)

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

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 \(y=2 x\)

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

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

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