Summary of Main Ideas

University:
Massachusetts Institute of Technology
Course:
18.089 | Review of Mathematics
Academic year:

2023
Views:

41

Pages:

1
Author:

Sporkz

Summary of main ideas As we know, the gradient of a function is perpendicular to the curve at any point on the curve. Our boundary is a level surface, and so would be the gradient of its associated function. The second thing that we know is that the boundary vector is perpendicular to the gradient of f at the maximum point. So at this point, both of these vectors are perpendicular to the boundary, and that means that they have to be going in the same direction. So at a typical point, they're going in two different directions. When grad f and grad g are perpendicular to the boundary, they are both in the same direction. However, when grad f is not perpendicular to the boundary and grad g is perpendicular to the boundary, then it could be that they are both moving in the same direction or opposite directions. And what does it mean to be in the same direction or opposite directions? Then we can think of grad f as lambda times grad g. And lambda might be positive if they're going in the same direction, and it might be negative if they're going opposite directions. There are four steps to this thought process, each one of which we have practiced separately but that must now be combined.