MATHS CALCULUS EXPERTS ONLY

For each of the following differentiable functions of two variables listed below

f(x, y) = ex cos(y)

f(x, y) = ex sin(y)

f(x, y) = x2 −y2

f(x, y) = 2xy

f(x, y) = 1

2 ln(x2 + y2) (x, y) 6= (0, 0)

f(x, y) = arctan( y

x ) x > 0 .

your job is to:

1) Graph f. You may use a 3-d graphing package for help. For exam- ple, on a mac, try ”grapher”.

2) Plot the gradient field of f (at a reasonable number of well chosen points).

3) Sketch the level sets of f (for a reasonable number of levels).

4) Solve the gradient flow differential equation with initial point (xo, yo).

∇f(r(xo,yo)(t)) = r ′

(xo,yo) (t)

r(xo,yo)(0) = (xo, yo)

5) Sketch the gradient flow trajectory with some special choices of the ini- tial point. For example answer the question : “How long will the flow last?”

So, for each of the six functions above, you have five tasks, therefore you have 30 tasks to perform. A conscientious effort should take you about 15- 20 written pages of work. Use what we discussed in lecture to get started. When you write it up, remember to work it all out on your thinking note pad, and once you have understood it, rewrite it with great care and hand

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it in to your TA. Lastly, sketch the gradient flow lines and the level sets together, like we did in the lecture. Think carefully about how these two sets of curves should meet.