Fourth Exam

1. Find a parametrization for the curve.
   1. The circle of radius $3$ parallel to the $xy$ -plane, centered at the point $(0,0,2)$ .
   2. The line from $P_{0}=(1,-3,2)$ to $P_{1}=(4, 1, -3)$ .
2. The path of a particle is described by
   $$x=(t-1)^{2} , \quad y = 2 , \quad z = 2 t^{3} - 3 t^{2}$$
   Find the velocity $\mathbf{v}(t)$ and the speed $\vert\vert\mathbf{v}(t)\vert\vert$ . Also find any times at which the particle stops.
3. A particle moves with $x=4 \cos 3 t$ , $y = 4 \sin 3 t$ . Sketch the particle's path. Find the particle's position, velocity, and acceleration vectors at $t=0$ , and add these vectors to your sketch.

4. Find $\int_{C} \mathbf{F} \cdot d \mathbf{r}$ for the given $\mathbf{F}$ and $C$ .
   1. $\mathbf{F}=2 y \, \hat{\mathbf{i}} - (\sin y) \, \hat{\mathbf{j}}$ counterclockwise around the unit circle $C$ starting at the point $(1,0)$ .
   2. $\mathbf{F} = x^{3} \, \hat{\mathbf{i}} + y^{2} \, \hat{\mathbf{j}} + z \, \hat{\mathbf{k}}$ and $C$ is the line from the origin to the point $(2, 3, 4)$ .
5. Find $\int_{C} \mathbf{F} \cdot d \mathbf{r}$ for the given $\mathbf{F}$ and $C$ using the Fundamental Theorem of Line Integrals.
   1. $\mathbf{F} = 2 x \, \hat{\mathbf{i}} - 4 y \, \hat{\mathbf{j}} + (2z-3) \, \hat{\mathbf{k}}$ and $C$ is the line from $(1,1,1)$ to the point $(2, 3, -1)$ .
   2. $\mathbf{F} = y \sin(xy) \, \hat{\mathbf{i}} + x \sin(xy) \, \hat{\mathbf{j}}$ and $C$ is the parabola $y=2x^{2}$ from $(1,2)$ to the point $(3, 18)$ .

6. Decide if $\mathbf{F} = 2 x \cos(x^{2} + z^{2}) \, \hat{\mathbf{i}} + \sin (x^{2} + z^{2}) \, \hat{\mathbf{j}} + 2z \cos(x^{2}+ z^{2}) \, \hat{\mathbf{k}}$ is the gradient of a function $f$ . If so, find $f$ . If not, explain why not.
7. Use Green's Theorem to calculate the circulation of $\mathbf{F}=(2x^{2}+3y) \, \hat{\mathbf{i}} + (2x + 3y^{2}) \, \hat{\mathbf{j}}$ around the triangle with vertices $(2,0)$ , $(0,3)$ , $(-2,0)$ , oriented counterclockwise.