Gradient

We discussed the gradient of a function of two variables yesterday.

Definition:
Let f(x,y) have continuous partial derivatives. The gradient of f(x,y) is the vector:

Another notation is grad f(x,y).

Alternative Form of the Directional Derivative (Using gradient):
There is an alternate way to get the directional derivative using the gradient of a function. If f is a differentiable function of x and y, then the directional derivative of f in the direction of the unit vector u is:

Important Properties of the Gradient:

1. If the gradient of the function is 0, then the directional derivative is 0 for all u.

2. The direction of maximum increase of f is given by the gradient of the function. The maximum value of the directional derivative is the magnitude of the gradient of the function.

3. The direction of minimum increase of f is given by the negative gradient of the function. The minimum value of the directional derivative is the magnitude of the gradient of the function.