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