proof help needed?

Question

In a neighborhood of the point a, the level curve C ={x is a element of R^2 : f(x) = c } can be parameterized by a differentiable function g:(-e,e)->R^2 with g(0)=a use the Chain Rule to prove that gradient f(a) is orthogonal to the tangent vector to C at a
any help on solving this question is much appreciated!

Answer 1

On a surface, the gradient vector points in the direction in which the value increases most rapidly - that is the steepest direction.

The level curve is a curve of constant value (or, at least, the portion of such a curve in the neighborhood of interest) so the tangent to the level curve is in the direction in which the value changes least rapidly.

What you want to show is that these directions are perpendicular when the gradient is not zero.

For a simple model, think of the northern hemisphere of the Earth. Everywhere except at the true North Pole, the gradient will point along the line of longitude. The level curve is just the line of latitude. And, as you'd expect the two lines are always (except at the pole itself) perpendicular.

Now if they weren't perpendicular, then as you move in the direction of the tangent of the level curve, you'd also have a non-zero component of motion in the direction of the gradient, so the value as you move would not be essentially constant but would change more rapidly than it needs to.

(Recall that given two vectors, A and B, you can decompose A into a component perpendicular to B and a component parallel to B. A and B being orthogonal means the component of A parallel to B is the 0 vector.)

Now that you understand what the problem is asking, ...