Which of the following equations produces a surface that is NOT shown here?

Find a so that the point (3, a, 1) is on the tangent plane to  $z=e^{xy}-4x^2y+3y^2$  at (0,1,4).

Find the directional derivative of  $f\left(x,y\right)=\sqrt{4x^2+3y}$  at (2,3) in the direction of  $\overrightarrow{i}-2\overrightarrow{j}$  .

For the level surface $3y^2z+xz^2=10$  find  $2\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$  at (1,-1,2)

Find the minimum value of f(x, y) = 2x + 3y + 2 given that $2x^2+5xy+4y^2=28$  

Let $f\left(x,y\right)=\left(x^2+y^2\right)e^x$  . This function has

Let D be the region in the first quadrant between the circles $x^2+y^2=1$  and  $x^2+y^2=4$ . Evaluate the integral $\int\int_D\frac{\left(x^2y\right)}{\left(x^2+y^2\right)^{\frac{3}{2}}}dA$  

Which of the following integrals represents the volume of the solid in the first octant that is bounded on the side by the surface $x^2+y^2=4$  and on the top by the surface  $x^2+y^2+z^2=4$  ?

Convert the integral to cylindrical, then evaluate it: $\int_{-2}^2\int_0^{\sqrt{4-x^2}}\int_0^{\left(4-x^2-y^2\right)}15\sqrt{x^2+y^2}dzdydx$  

Compute  $\int\int\int_E\ z\ dV$ , where E is bounded by the sphere  $x^2+y^2+z^2=1$  and the coordinate planes in the first octant.

A particle is travelling on the path y = x from (0, 0) to (1, 1). For which of the following force vector fields is the work done equal to 0?

If $\overrightarrow{F}=\left(3+2xy\right)\overrightarrow{i}+\left(x^2-3y^2\right)\overrightarrow{j}$ and  $\overrightarrow{F}=\overrightarrow{\nabla}f$  , find  $\int_C\overrightarrow{\nabla}f.d\overrightarrow{r}$   if the curve C is parametrized as  $\overrightarrow{r}\left(t\right)=e^t\sin\left(t\right)\overrightarrow{i}+e^t\cos\left(t\right)\overrightarrow{j},\ 0\le t\le\pi$  

According to Green’s Theorem, which of the following line integrals is NOT equal to the area of the region enclosed by a simple curve C?

Find grad(div(F)) · curl(F) for  $F\left(x,y,z\right)=xy\overrightarrow{i}+yz\overrightarrow{j}+xz\overrightarrow{k}$ at (1,-1,2)

Find $\int_C\left(x+y^3e^y\right)dy-2ydx$  where C goes clockwise around the trapezoid with corners (0, 0), (0, 4), (2, 1), (2, 3). 

Find the surface area of the surface with parametric equations x = u + v, y = u − v, z = 2v, 0 ≤ u ≤ 1, 0 ≤ v ≤ 1.

If S is that part of the paraboloid $z=x^2+y^2$  with z ≤ 4, and  $\overrightarrow{n}$  is the downward pointing unit normal, and  $\overrightarrow{F}\left(x,y,z\right)=x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k}$ , then  $\int\int_S\overrightarrow{F}.\overrightarrow{n}dS=$  

Evaluate the integral $\int\int_S\text{curl}\overrightarrow{F}.d\overrightarrow{S}$ using Stoke’s Theorem, where  $\overrightarrow{F}=-y\overrightarrow{i}+x\overrightarrow{j}+xyz\overrightarrow{k}$  and S is the part of the sphere  $x^2+y^2+z^2=4$  that lies above the xy-planes, oriented upward.

Let $\overrightarrow{F}=\left(xy^2+1,yz^2-x,zx^2+y\right)$ . Use the Divergence Theorem to evaluate  $\int\int_S\overrightarrow{F}.d\overrightarrow{S}$  where where S is the boundary surface of the solid  $E=\left\{(x,y,z)|x^2+y^2+z^2=4,x≥0,y≥0,z≥0\right\}$  with an outward orientation.