91Ó°ÊÓ

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

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

Use Green's Theorem to find the counterclockwise circulation and outward flux for the field \(\mathbf{F}\) and curve \(C\). \(\mathbf{F}=\left(y^{2}-x^{2}\right) \mathbf{i}+\left(x^{2}+y^{2}\right) \mathbf{j}\) C: The triangle bounded by \(y=0, x=3,\) and \(y=x\)

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).

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

Use the Divergence Theorem to find the outward flux of \(\mathbf{F}\) across the boundary of the region \(D\) Sphere \(\quad \mathbf{F}=x^{3} \mathbf{i}+y^{3} \mathbf{j}+z^{3} \mathbf{k}\) \(D:\) The solid sphere \(x^{2}+y^{2}+z^{2} \leq a^{2}\)

Use the surface integral in Stokes" Theorem to calculate the flux of the curl of the field \(\mathbf{F}\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\). $$\begin{aligned} &\mathbf{F}=2 \mathbf{i}+3 x \mathbf{j}+5 y \mathbf{k}\\\ &S: \quad \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+\left(4-r^{2}\right) \mathbf{k}\\\ &0 \leq r \leq 2, \quad 0 \leq \theta \leq 2 \pi \end{aligned}$$

Integrate \(G(x, y, z)=x+y+z\) over the portion of the plane \(2 x+2 y+z=2\) that lies in the first octant.

In Exercises \(13-16\), find the line integrals along the given path \(C\). $$\int_{c}(x-y) d x, \text { where } C: x=t, y=2 t+1, \text { for } 0 \leq t \leq 3$$

Use Green's Theorem to find the counterclockwise circulation and outward flux for the field \(\mathbf{F}\) and curve \(C\). \(\mathbf{F}=\left(\tan ^{-1} \frac{y}{x}\right) \mathbf{i}+\ln \left(x^{2}+y^{2}\right) \mathbf{j}\) C: The boundary of the region defined by the polar coordinate inequalities \(1 \leq r \leq 2,0 \leq \theta \leq \pi\)