91Ó°ÊÓ

Sketch the level curve or surface passing through the indicated point. Sketch the gradient at the point. $$ f(x, y)=y-x^{2} ;(2,5) $$

Graph the curve traced by the given vector function. $$ \mathbf{r}(t)=t \mathbf{i}+2 t \mathbf{j}+\cos t \mathbf{k} ; t \geq 0 $$

Show that the given line integral is independent of the path. Evaluate in two ways: (a) Find a potential function \(\phi\) and then use Theorem 9.9.1, and (b) Use any convenient path between the endpoints of the path. $$ \int_{(1,0)}^{(3,2)}(x+2 y) d x+(2 x-y) d y $$

Verify Stokes' thearem. Assume that the surface \(S\) is orienled upwand. \(\mathbf{F}=z \mathbf{i}+x \mathbf{j}+y \mathbf{k} ; S\) that portion of theplane \(2 x+y+2 z=6\) in the first octant

Use a double integral in polar coordinates to find the area of the region bounded by the graphs of the given polar equations. $$ 2 \sin \theta, r \quad 1, \text { common area } $$

Fill in the blank or answer true/false. Where appropriate, assume continuity of \(P, O\), and their first partial derivatives. The binormal vector is perpendicular to the osculating plane.____

Compute the gradient for the given function. $$ F(x, y, z)=\frac{x y^{2}}{z^{3}} $$

In Problems \(1-6\), graph some representative vectors in the given vector field. $$ \mathbf{F}(x, y)=y \mathbf{i}+x \mathbf{j} $$

Graph some representative vectors in the given vector field. $$ \mathbf{F}(x, y)=y \mathbf{i}+x \mathbf{j} $$

Find the surface area of that portion of the cylinder \(x^{2}+z^{2}=16\) that is above the region in the first quadrant bounded on the graphs of \(x=0, x=2, y=0, y=5\)