Due to the amount of confusion and the large number of emails, I have written up the solution to Problem 1 from Test 2.

**The Problem**

*Determine the path of steepest descent along the surface from the point *

There are a few things to note – the first thing we must do is find which direction points ‘downwards’ the most. So we note that for a function we know that points ‘upwards’ the most at all points where it isn’t zero. So at any point we go in the direction

The second thing to note is that we seek a path, not a direction. So let us take a curve that parametrizes our path:

So

As the velocity of the curve points in the direction of the curve, our path satisfies:

These are two ODEs that we can solve by separation of variables (something that is, in theory, taught in 1502 – for more details, look at chapter 9 in Salas, Hille, and Etgen). Let’s solve the y one:

for a constant k

for a constant A

for a new constant A

Solving both yields:

Now let’s get rid of the t. Note that and . Using these together, we can get rid of t by noting that Rewriting, we get

So the path is given by

Good luck on your next test!

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I have been informed that I had a small typo (two actually, and they cancelled each other out). and , but dividing cancels them out. Also note that anywhere, so we don’t have to worry about that detail.

Could you please elaborate on the treatment of constant (k and A)? I do not get how you changed the constant ‘k’ to ‘A’…thanks

Sure. So we had , and I think the question refers to how this becomes .

We have the following: . Now is just some constant, and if we call it A we see that .