What follows are extra problems for Homework #7.
I added a small extra problem here.
Take a look at this 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 ).
b) Give an equation for a closed (i.e. closed loop), non-trivial (i.e. not just a point) curve for which .
c) Give an equation for a curve for which .
d) Give an equation for a curve for which .
Take a look at this vector field . 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 for which .
b) On this vector field, draw a closed non-trivial loop for which . Choose a which does not intersect .
c) On this vector field, draw two different points and and two different curves and such that the following three things hold:
1) is a curve with endpoints and and orientation going from to ;
2) is a curve with endpoints and and orientation going from to ;
Let be a constant vector field in the plane (2D vector field). Is there a closed loop in this field for which ? 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.