13.6 Day 2 Gradients For an animation of this concept visit:

Diagram demonstrating gradients

Gradients are perpendicular to the level curves Here is another website that demonstrates directional derivatives and gradients:

For two animations that demonstrates the gradient visit eadings/directderiv/

Example 3 Find the gradf(x,y) at the point (1,2)

Note: this is the method that we will use most often to find a directional derivative

Example 4 Find the directional derivative of f(x,y) at (-3/4,0) in the direction from P(-3/4,0) to Q(0,1)

Example 4 solution

The proof of these properties is in the book on page 936

Example 5 Find the direction of maximum increase in degrees Celsius on the surface of a metal plate. What is the rate of increase? (see diagram on next slide)

Solution to example 5

This shows the direction of maximum increase for example 5

Example 6 A heat seeking particle is located at the point (2,-3) on a metal plate whose temperature is given by Find the path of the particle as it continuously moves in the direction of maximum temperature increase. (see next slide for diagram)

Diagram for the solution of example 6

Gradient Is Normal to Level Curves

Example 7 Sketch the level curve corresponding to c = 0 for f(x,y) and find a normal vector at several points on the curve. (diagram is on the next slide)

Diagrams for Example 7.

Example 8 Find the gradient for f(x,y,z). Find the direction of maximum increase at the point (2, -1, 1)

Example 8 Solution

Q: What do you get when you multiply Santa Claus times i? A: He becomes real. Note: it is better to multiply him by – i or he will be negative.