## Fall 2010 UBC

### Homework #7 Extra Problems

What follows are extra problems for Homework #7.

## Bonus Problem.

I added a small extra problem here.

## Problem 1.

Take a look at this vector field $\mathbf{F}$.

For each part of this problem, please provide a justification (it need not be a calculation:  it can be an intuitive explanation).

a) Identify this vector field (i.e. give a likely formula for $\mathbf{F}$).

b) Give an equation for a closed (i.e. closed loop), non-trivial (i.e. not just a point) curve $C$ for which $\int_C \mathbf{F} \cdot d\mathbf{r} = 0$.

c) Give an equation for a curve $C$ for which $\int_C \mathbf{F} \cdot d\mathbf{r} > 0$.

d) Give an equation for a curve $C$ for which $\int_C \mathbf{F} \cdot d\mathbf{r} < 0$.

## Problem 2.

Take a look at this vector field $\mathbf{F}$.  I don’t expect you to be able to figure out what vector field it is (i.e. what equation gives it):

Either print out this vector field or draw a rough copy of it in your notes (showing the main features but with far fewer vectors; don’t waste too much time).

a) On this vector field, draw a closed non-trivial loop $C_1$ for which $\int_{C_1} \mathbf{F} \cdot d\mathbf{r} > 0$.

b) On this vector field, draw a closed non-trivial loop $C_2$ for which $\int_{C_2} \mathbf{F} \cdot d\mathbf{r} < 0$.  Choose a $C_2$ which does not intersect $C_1$.

c) On this vector field, draw two different points $P$ and $Q$ and two different curves $C_3$ and $C_4$ such that the following three things hold:

1) $C_3$ is a curve with endpoints $P$ and $Q$ and orientation going from $P$ to $Q$;

2) $C_4$ is a curve with endpoints $P$ and $Q$ and orientation going from $P$ to $Q$;

3) $\int_{C_3} \mathbf{F} \cdot d\mathbf{r} < \int_{C_4} \mathbf{F} \cdot d\mathbf{r}$.

## Problem 3.

Let $\mathbf{F}$ be a constant vector field in the plane (2D vector field).  Is there a closed loop $C$ in this field for which $\int_C \mathbf{F} \cdot d\mathbf{r} \neq 0$?  Why or why not?  (If there is, write it down.  If there is not, prove there is not.)  NOTE:  If the answer is a proof, then I would like a direct proof, not one based on the Fundamental Theorem of Line Integrals or anything from section 17.3.  This is a simple example where you can check it out directly.