MATH 20C Lecture 17

14.5 Gradient and Directional Derivatives

Volume of a Rectangular Cuboid

Suppose is computed using
Use the linear approx. to estimate the maximal percentage error in if each of these values has a possible percentage error of at most

Sol:


Maximal percentage erros in are 2%



Maximal error
Maximal percentage error

14.5

Recall that
rate of change in the x-direction
rate of change in the y-direction

Gradient

The gradient (vector) of at is

• and define
  sometimes we omit reference to the point
  and
  or

Properties of Gradient

differentiable (in 2 or 3 variables)
c constant, scalar valued
Then

4. Chain Rule
   differentiable in , then
   or
5. Chain rule for path
   If and are differentiable

Let be a unit vector.
How do we measure the rate of change in the direction of ?

Directional Derivative

The directional derivative of at in the direction of the unit vector is

$D_{\vec{u}}f(P)=\nabla f_P\cdot \vec{u}$