8—Multivariable Calculus 1868.5 Gradient
The equation (8.13) for the differential has another geometric interpretation. For a function such as
f(x,y) =x^2 + 4y^2 , the equations representing constant values offdescribe curves in thex-yplane.
In this example, they are ellipses. If you start from any fixed point in the plane and start to move away
from it, the rate at which the value offchanges will depend on the direction in which you move. If you
move along the curve defined byf=constant thenfwon’t change at all. If you move perpendicular
to that direction thenfmay change a lot.
The gradient of f at a point is the vector pointing in
the direction in which f is increasing most rapidly, and
the component of the gradient along that direction is thederivative offwith respect to the distance in that direction.
To relate this to the partial derivatives that we’ve been using, and to understand how to compute
and to use the gradient, return to Eq. (8.13) and write it in vector form. Use the common notation for
the basis:ˆxandyˆ. Then let
d~r=dxxˆ+dyyˆ and G~=
(
∂f
∂x
)
yxˆ+
(
∂f
∂y
)
xyˆ (8.15)
The equation for the differential is now
df=df(x,y,dx,dy) =G~.d~r (8.16)
G~
d~r
θ
Because you know the properties of the dot product, you know that this isGdrcosθand it is
largest when the directions ofd~rand ofG~are the same. It’s zero when they are perpendicular. You
also know thatdfis zero whend~ris in the direction along the curve wherefis constant. The vector
G~ is therefore perpendicular to this curve. It is in the direction in whichf is changing most rapidly.
Also becausedf=Gdrcos 0, you see thatGis the derivative offwith respect to distance along that
direction.G~is the gradient.
For the examplef(x,y) =x^2 + 4y^2 ,G~= 2xˆx+ 8yˆy. At each point in thex-yplane it provides
a vector showing the steepness offat that point and the direction in whichfis changing most rapidly.
Notice that the gradient vectors are twice as long where the ellipses are closest together as
they are at the ends where the ellipses are farthest apart. The function changes more rapidly in the