Identify the surface defined by the equation  $x^2+y^2+2z-z^2=0$  

The vector field  $F\left(x,y\right)=\left\{2xe^y+1,x^2e^y\right\}$   is conservative. Compute the work done by the field in moving an object along the path C:  $r\left(t\right)=\left\{\cos t,\sin t\right\},\ 0\le t\le\ \pi$  

Compute $\int_C\left(e^{2x}+y^2\right)dx+\left(14xy+y^2\right)dy$  where C is the boundary of the region bounded by the y-axis and the curve  $x=y-y^2$   oriented counterclockwise.

Find the linear approximation of  $f\left(x,y\right)=y\sqrt{x}$ at (4, 1).

Compute $curl\ F\left(\pi,1,1\right)$ where  $F=\left\{x+y,yz,\sin\left(x\right)\right\}$   

If $f_{xy}=x\sin\left(xy^2\right)$  , compute  $f_{yx}\left(\pi,1\right)$  

Find the direction in which $f\left(x,y,z\right)=\frac{x}{z}-yz$  decreases most rapidly at the point (4,1,1).

Let M and m denote the maximum and the minimum values of  $f\left(x,y\right)=x^2-2x+y^2+3$  in the disk  $x^2+y^2\le1$  . Find M + m. 

Evaluate the integral $\int\int_D2\pi\sin\left(x^2\right)dA$  where D is the region in the x-y plane bounded by the lines y = 0, y = x and  $x=\sqrt{\pi}$  .

Evaluate the double integral  $\int\int_D\ 2e^{\left(x^2+y^2\right)}dA$  where D is the region bounded by the x-axis and the curve  $y=\sqrt{1-x^2}$  

Compute the triple integral  $\int\int\int_E\ 3y\ dV$  where E is a region under the plane x + y + z = 2 in the first octant.

 The integral $\int_0^{\sqrt{2}}\int_{-\sqrt{2-x^2}}^{\sqrt{2-x^2}}\int_{\sqrt{3x^2+3y^2}}^{\sqrt{8-x^2-y^2}}xy^2z\ dzdydx$ when converted to cylindrical coordinates becomes

Convert the integral to spherical coordinates and evaluate it:  $\int_{-2}^2\int_0^{\sqrt{4-x^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{8-x^2+y^2}}3\ dzdydx$  

Compute the line integral  $\int_C\ F.dr$  where  $F=\left\{xy,x+y\right\}$  and C is the curve  $y=x^2$  from (0,0) to (1,1)

 Let S be the part of the surface z = xy + 1 that lies within the cylinder  $x^2+y^2=1$  . Find the area of the surface S. 

Find the surface area of the parametric surface  $r\left(u,v\right)=\left(u^2,uv,\frac{v^2}{2}\right)$  with  $0\le u\le3,\ 0\le v\le1.$  

Use Stokes’ Theorem to evaluate the integral $\int_C\ y\ dx+z\ dy+x\ dz$  , where C is the intersection of the surfaces  $x^2+y^2=1$  and x + y + z = 5. C is oriented counterclockwise when viewed from above.

Evaluate the flux integral $\int\int_S\ F.\ dS$  where  $F\left(x,y,z\right)=\left\{3\cos y,x\cos z,z^3\right\}$  and S complete boundary surface of the solid region bounded by the cylinder  $y^2+x^2=2$  and the planes x = 1 and x = 3. S is oriented by the outward normal. 

The position function of a Space Shuttle is $r\left(t\right)=\left\{t^2,-t,6\right\},\ t\ge0$ . The International Space Station has coordinates (16, −5, 6). In order to dock the Space Shuttle with the Space Station the captain plans to turn off the engine so that the Space Shuttle coasts into the Space Station. At what time should the captain turn off the engines? Assume there are no other forces acting on the Space Shuttle other than the force of the engine.

The area of the triangle with vertices (2, 1, 1), (1, 2, 1), (1, 1, 2) is

The arclength of the curve  $r\left(t\right)=\left(2t\right)i+\left(t^2\right)j+\left(\ln t\right)k$  for  $1\le t\le2$  is

 A particle has position r(t) with acceleration $a\left(t\right)=\left(t\right)i+\left(3t^2\right)k$ and the initial conditions  $v\left(0\right)=i+j+k$  and  $r\left(0\right)=0$  . Then  $r\left(1\right)$  is

A continuous function f(x, y) defined on the region D = [1, 3] × [0, 1] has its absolute minimum value equal to 4 and its absolute maximum value equal to 5. Which of the following numbers could equal $\int\int_D\ f\left(x,y\right)dA$ ? 

Suppose that z is defined as a function of x and y by the equation cos(xyz) = x + 3y + 2z. Use implicit differentiation to find the value of $\frac{\partial z}{\partial y}\left(0,1\right).$  

Consider the tangent plane to the surface z = ln(x − 4y) at the point (9, 2, 0). This tangent plane also contains the point (2, 1, λ). Find λ.

Find the directional derivative of $f\left(x,y\right)=xe^{y^2}+e^{x+y}$  at the point (0, 0) in the direction of the vector  $3i+4j$  

Suppose E is the region bounded above by the cylinder $x^2+z^2=5$ , below by the plane z = 1, and on the sides by the planes y = −1 and y = 2. Find $\int\int\int_E\ z\ dV$  

The points P = (0, 1) and  $Q=\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$  are critical points of the function  $f\left(x,y\right)=2x^3-3x^2y-y^3+3y$ . Classify each as a relative maximum, relative minimum, or saddle point.

A lamina with density ρ(x, y) = xy occupies the region of the plane bounded by $y=x^2$ , y = 1 and x = 0. The mass of the lamina is equal to  $\frac{1}{6}$  . Find the y-coordinate of its center of mass.

Find the surface area of the part of the paraboloid  $z=\frac{x^2}{2}+\frac{y^2}{2}$  that lies between the cylinders  $x^2+y^2=8$  and  $x^2+y^2=24$  

Evaluate the line integral  $\int_C\ \overrightarrow{F}.\ d\overrightarrow{r}$  where  $\overrightarrow{F}\left(x,y,z\right)=y\overrightarrow{i}-x\overrightarrow{j}+xy\overrightarrow{k}$  and C is parameterised by  $\overrightarrow{r}\left(t\right)=\left(\sin t\right)\overrightarrow{i}+\left(\cos\ t\right)\overrightarrow{j}+t\overrightarrow{k}$  

 Use Green’s Theorem to evaluate $\int_C\ x^2\ dy$ where C is the boundary of the rectangle with vertices {(0, 0),(2, 0),(2, 3),(0, 3)}, oriented counterclockwise.

If $f\left(x,y,z\right)=x^2yz-xy^2+2xz^2$ , then  $div(grad(f))$  at (1,1,1) is equal to

 Let S be the parametric surface $\overrightarrow{r}\left(u,v\right)=\left(v\cos u\right)\overrightarrow{i}+\left(v\sin u\right)\overrightarrow{j}+\left(2v^2\right)\overrightarrow{k}$ with (u, v) in [0, 2] × [0, 2]. Then S is part of a

Find the surface area of the parametric surface $\overrightarrow{r}\left(u,v\right)=\left(u+v\right)\overrightarrow{i}+v\overrightarrow{j}+u\overrightarrow{k}$  with (u, v) in  $\left[0,\pi\right]\times\left[0,\sqrt{3}\right]$  .

Let S be the part of the sphere  $x^2+y^2+z^2=1$  above the plane  $z=\frac{1}{2}$  . Compute the surface integral $\int\int_S\ 12z^2\ dS$  

The flux of the vector field $\overrightarrow{F}\left(x,y,z\right)=x\overrightarrow{i}+\left(x+y\right)\overrightarrow{j}+z\overrightarrow{k}$ across the surface of the plane x + y + z = 1 in the first octant, oriented upward, is equal to:

 Let S be the part of the circular paraboloid $z=x^2+y^2$ below the plane z = 4 with upward orientation. Let  $\overrightarrow{F}\left(x,y,z\right)=xz\overrightarrow{j}+yz\overrightarrow{k}$ . Compute  $\int\int_S\ curl\ \left(\overrightarrow{F}\right).\ \overrightarrow{n}\ dS$ .

Hint: You may need to use one or both of these integrals:

 $\int_0^{2\pi}\left(\cos^2t\right)dt=\int_0^{2\pi}\left(\sin^2t\right)dt=\pi$  

Suppose  $\overrightarrow{F}\left(x,y,z\right)=2xy^2\ \overrightarrow{i}+2yx^2\ \overrightarrow{j}-\left(x^2+y^2\right)z\ \overrightarrow{k}$   and S is the boundary surface of the solid enclosed by the cylinder $x^2+y^2=1$   and the planes z = −1 and z = 1. S is a closed surface oriented by the outward normal. Calculate the flux integral  $\int\int_S\ \overrightarrow{F}.\ d\overrightarrow{S}$  

Which of the following pairs of planes are orthogonal to each other?

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.