WOLFRAM|DEMONSTRATIONS PROJECT

Gradients in 2D and 3D

f(x,y) =

( x 0 , y 0 ) =

For a smooth surface

S

in 3D, representing a function

z=f(x,y)

, the gradient at a point

(

x

0

,

y

0

)

on

S

is a vector in the direction of maximum change of

f(x,y)

. Also shown is the corresponding contour plot, which is the projection of

S

onto the

x

-

y

plane. The red arrows on the surface and contour plots show the magnitude and direction of the gradient.

The dot shows the point

(

x

0

,

y

0

)

at which the gradient is computed. You can vary the point by dragging the locator.

Change the function with the pull-down menu.

Rotate the graph to convince yourself that the gradient is normal to the surface at the point.