Unit 5 test: Vector Calculus

Let $\mathbf{F}(x, y) = \langle x - y,\ xy\rangle$. Find $\mathbf{F}(2, 3)$.

Evaluate $\displaystyle\int_C (x + 2y)\,ds$, where $C$ is the segment from $(0, 0)$ to $(6, 8)$.

Find the work done by $\mathbf{F}(x, y) = \langle -y,\ x\rangle$ on a particle moving counterclockwise along the circle $x^2 + y^2 = 4$ from $(2, 0)$ to $(0, 2)$.

Let $\mathbf{F}(x, y) = \langle 2xy,\ x^2 + 3y^2\rangle$. Evaluate $\displaystyle\int_C \mathbf{F}\cdot d\mathbf{r}$ along any path from $(1, 1)$ to $(2, -1)$.

Which vector field on $\mathbb{R}^3$ is conservative?

Use Green's theorem to evaluate $\displaystyle\oint_C (x^2 - 2y)\,dx + (4x + y^3)\,dy$, where $C$ is the circle $x^2 + y^2 = 9$, counterclockwise.

Evaluate $\displaystyle\oint_C x\,dy$, where $C$ is the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1$, counterclockwise.

Find $\operatorname{div}\mathbf{F}$ for $\mathbf{F} = \langle xy,\ yz,\ zx\rangle$.

Let $\mathbf{F} = \langle z,\ x^2,\ y\rangle$. Find $\operatorname{curl}\mathbf{F}$ at $(3, 0, 0)$.

Which statement is true for every function $f$ with continuous second partial derivatives?

Find the area of the part of the plane $z = 2x + 2y$ that lies above the unit disk $x^2 + y^2 \le 1$.

Find the flux of $\mathbf{F} = \langle 0, 0, z\rangle$ upward through the part of $z = 1 - x^2 - y^2$ above the $xy$-plane.

Let $\mathbf{F} = \langle -y,\ x,\ z\rangle$ and let $S$ be the part of $z = 9 - x^2 - y^2$ with $z \ge 0$, oriented upward. Use Stokes' theorem to evaluate $\displaystyle\iint_S \operatorname{curl}\mathbf{F}\cdot d\mathbf{S}$.

Use the divergence theorem to find the outward flux of $\mathbf{F} = \langle x^3,\ y^3,\ z^3\rangle$ across the sphere $x^2 + y^2 + z^2 = 4$.

Find the outward flux of $\mathbf{F} = \langle x,\ y^2,\ z^3\rangle$ across the boundary of the cube $0 \le x, y, z \le 1$.