Exercises

01

Computing scalar curl

Compute $\nabla {\times }_z\mathbf{F}$, where $\mathbf{F}=(e^{x+y},e^{x-y})$.

02

Computing scalar curl

Compute $\nabla {\times }_z\mathbf{F}$, where $\mathbf{F}=\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$.

03

Understanding curls

• (a) Draw the vector field $\mathbf{F}=(-y,x)$.
• (b) Compute the curl $\nabla {\times }_z\mathbf{F}$, where $\mathbf{F}=(-y,x)$.
• (b) Draw the vector field from 12A-02.

04

Line integral and vector field

What is the vector field being integrated in this line integral?

$$\oint x^{3}dy-e^{y}dx$$

05

Circulation calculated around circles

Repeat the reasoning above but going around a small circle instead of a box.
You can use the single parametrization $\mathbf{r}(t)=(a+r\cos (t),b+r\sin (t))$ for $r$ a small constant, and $(a,b)$ the center point of the circle.

The derivation is faster with this method, but circles do not tile the plane the way boxes do, and we need a tiling of the plane to prove Green’s Theorem.