91Ó°ÊÓ

What does it mean if the curl of a vector field is zero throughout a region?

Sketch a two-dimensional vector ficld that has zero curl everywhere in the plane.

Describe the usual orientation of a closed surface such as a sphere.

Give three equivalent properties of conservative vector fields.

Sketch the following vector fields. $$\mathbf{F}=\langle x,-y\rangle$$

Sketch the following vector fields. $$\mathbf{F}=\left\langle\frac{x}{\sqrt{x^{2}+y^{2}}}, \frac{y}{\sqrt{x^{2}+y^{2}}}\right\rangle$$

Area of regions Use a line integral on the boundary to find the area of the following regions. A disk of radius 5

For the vector field \(\mathbf{F}\) and curve \(C\), complete the following: a. Determine the points (if any) along the curve C at which the vector field \(\mathbf{F}\) is tangent to \(C\). b. Determine the points (if any) along the curve C at which the vector field \(\mathbf{F}\) is normal to \(C\) c. Sketch \(C\) and a few representative vectors of \(\mathbf{F}\) on \(C\). $$\mathbf{F}=\left\langle\frac{1}{2}, 0\right\rangle ; C=\left\\{(x, y): y-x^{2}=1\right\\}$$

Area of regions Use a line integral on the boundary to find the area of the following regions. The region bounded by the parabolas \(\mathbf{r}(t)=\left\langle t, 2 t^{2}\right\rangle\) and \(\mathbf{r}(t)=\left\langle t, 12-t^{2}\right\rangle,\) for \(-2 \leq t \leq 2\)

Curl of a vector field Compute the curl of the following vector fields. $$\mathbf{F}=\left\langle x^{2}-y^{2}, x y, z\right\rangle$$