Gradients and Directional Derivatives

Chp 15.6

Putting it all together…

• Over the past several classes you have learned how to use and take partial derivatives

• Today we look at the essential difference between defining slope along a 2D curve and what it means on a 3D surface or curve

2

2 2

10( , )

1

xe yf x y

x y

Example…What’s the slope of at (0,1/2)?

What’s wrong with the way the question is posed?

What’s the slope along the direction of the x-axis?

What’s the slope along the direction of the y-axis?

Quick estimate from the contour plot:

(0 4) 4

(0.5 0) 0.5

8

z

y

m

z

Look at this in Excel:

The Directional Derivative

• We need to specify the direction in which the change occurs…

• Define, via a slightly modified Newton quotient:

• This specifies the change in the direction of the vector u = <a,b>

( , ) ( , ) ( , )u x yD f x y f x y a f x y b

The Gradient

• We can write the Directional derivative as:

1 2 3( , , ) , , , ,u x y zD f x y z f f f u u u

Gradient of f(x,y,z)

( , , )f x y z

A Key Theorem

• Pg 982 – the Directional derivative is maximum when it is in the same direction as the gradient vector!

• Example: If your ski begins to slide down a ski slope, it will trace out the gradient for that surface!