Exercises

01

Gradient fields are conservative

Suppose $\mathbf{F}=\nabla f$ for some scalar field $f$. Show that $\nabla {\times }_z\mathbf{F}=0$.

02

(not for presentation)

Conservative fields are gradients

Suppose that $\nabla {\times }_z\mathbf{F}=0$. Find a scalar field $f$ such that $\mathbf{F}=\nabla f$. (Hint: define it using line integrals along paths!)

03

Gradient of angle

Let $f(x,y)=\theta ={\tan }^{-1}(\frac{y}{x})$. Show that:

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

We saw above that $\nabla {\times }_z\mathbf{F}=0$ for $(x,y)\ne (0,0)$. Compute the integral ${\int }_{{\mathcal{C}}_n}\mathbf{F}\cdot d\mathbf{s}$, where ${\mathcal{C}}_n$ is a path that winds $n$-times around a circle of radius $R$. (For example, $\mathbf{r}(t)=(R\cos (t),R\sin (t))$ with $t\in [0,2\pi n]$.)

Notice that $\mathbf{F}$ is not defined at $(x,y)$. So the applicable domain for Green’s Theorem has a ‘hole’ containing $(0,0)$.