91Ó°ÊÓ

Work Find the work done by the gradient of \(f(x, y)=(x+y)^{2}\) counterclockwise around the circle \(x^{2}+y^{2}=4\) from (2,0) to itself.

Circulation and flux Find the circulation and flux of the fields $$ \mathbf{F}_{1}=x \mathbf{i}+y \mathbf{j} \quad \text { and } \quad \mathbf{F}_{2}=-y \mathbf{i}+x \mathbf{j} $$ around and across each of the following curves. a. The circle \(r(t)=(\cos t) i+(\sin t)\\}, \quad 0 \leq t \leq 2 \pi\) b. The ellipse \(r(t)=(\cos t) i+(4 \sin t) j, \quad 0 \leq t \leq 2 \pi\)

Find the circulation and flux of the field \(\mathbf{F}\) around and across the closed semicircular path that consists of the semicircular arch \(\mathbf{r}_{1}(t)=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq \pi,\) followed by the line segment \(\mathbf{r}_{2}(t)=r \mathbf{i},-a \leq t \leq a\) $$\mathbf{F}=x \mathbf{i}+y \mathbf{j}$$

Find the circulation and flux of the field \(\mathbf{F}\) around and across the closed semicircular path that consists of the semicircular arch \(\mathbf{r}_{1}(t)=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq \pi,\) followed by the line segment \(\mathbf{r}_{2}(t)=r \mathbf{i},-a \leq t \leq a\) $$\mathbf{F}=-y^{2} \mathbf{i}+x^{2} \mathbf{j}$$

a. Find a parametrization for the hyperboloid of one sheet \(x^{2}+y^{2}-z^{2}=1\) in terms of the angle \(\theta\) associated with the circle \(x^{2}+y^{2}=r^{2}\) and the hyperbolic parameter \(u\) associated with the hyperbolic function \(r^{2}-z^{2}=1 .\) (Hint: \(\left.\cosh ^{2} u-\sinh ^{2} u=1 .\right)\) b. Generalize the result in part (a) to the hyperboloid \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)-\left(z^{2} / c^{2}\right)=1\)

You have been asked to find the path along which a force field \(\mathbf{F}\) will perform the least work in moving a particle between two locations. A quick calculation on your part shows \(\mathbf{F}\) to be conservative. How should you respond? Give reasons for your answer.

Flow integrals Find the flow of the velocity field \(\mathbf{F}=\) \((x+y) \mathbf{i}-\left(x^{2}+y^{2}\right) \mathbf{j}\) along each of the following paths from (1,0) to (-1,0) in the \(x y\) -plane. a. The upper half of the circle \(x^{2}+y^{2}=1\) b. The line segment from (1,0) to (-1,0) c. The line segment from (1,0) to (0,-1) followed by the line segment from (0,-1) to (-1,0)

Use Equation (7) to find the surface integral of the field \(\mathbf{F}\) over the portion of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) in the first octant in the direction away from the origin. $$\mathbf{F}(x, y, z)=\frac{x \mathbf{i}+y \mathbf{j}+z \mathbf{k}}{\sqrt{x^{2}+y^{2}+z^{2}}}$$

Find the area of the surface cut from the paraboloid \(x^{2}+y^{2}-z=\) 0 by the plane \(z=2\)

Find the flux of the field \(\mathbf{F}(x, y, z)=4 x \mathbf{i}+4 y \mathbf{j}+2 \mathbf{k}\) outward (away from the \(z\) -axis) through the surface cut from the bottom of the paraboloid \(z=x^{2}+y^{2}\) by the plane \(z=1\).