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}.

a vector field

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):

another vector field

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.

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