Using Parameterizations to Calculate Line Integrals

Is $\vF=\langle xy,y^2\rangle$ a gradient vector field? Why or why not?

1. Find the work done by the vector field $\vF(x,y,z) = 6x^2z\vi + 3y^2\vj + x\vk$ on a particle that moves from the point $(3,0,0)$ to the point $(3,0,6\pi)$ along the helix given by $\vr(t) = \langle 3\cos(t),3\sin(t),t\rangle$.

2. Let $\vF(x,y) = \langle 0,x\rangle$. Let $C$ be the closed curve consisting of the top half of the circle of radius $2$ centered at the origin and the portion of the $x$-axis from $(2,0)$ to $(-2,0)$, oriented clockwise. Find the circulation of $\vF$ around $C$.

1. Compute $\int_{C_1} \vF\cdot d\vr$ when $\vF=\langle x^2,xy\rangle$ and $C_1$ is the line segment from $(0,0)$ to $(2,2)$.

2. Compute $\int_{C_2} \vF\cdot d\vr$ when $\vF=\langle x^2,xy\rangle$ and $C_2$ is the line segment from $(2,2)$ to $(0,0)$.

If the wind in a region of space is given by $\vF=\langle y+z,z-x,-z \rangle$ and a helicopter flies along the path given by $\vr(t) = \langle 10 \sin(t),10\cos(t),(10-t)^2 \rangle$ as $0\leq t\leq4\pi$. Calculate the work done by the wind on the helicopter.

Let $C_3$ be the circle of radius $7$ centered at the origin traveled counterclockwise. Compute $\int_{C_3} \langle M, N\rangle\cdot d\vr$ when:

1. $\langle M,N\rangle = \langle x,y \rangle$

2. $\langle M,N\rangle = \langle -y,x \rangle$

3. $\langle M,N\rangle = \langle 3,x \rangle$