The Directional Derivative (continued)

Suppose z = f(x,y) represents a surface in xyz. Further suppose that the surface

F(x,y,z) = z – f(x,y) is continuously differentiable. Then grad F is a vector normal

to the surface. Call this normal vector n .

n = ∂F/∂x i + ∂F/∂y j + ∂F/∂z k = grad F

Here n is normal to the tangent plane at each point of the surface F(x, y, z)

So grad F can be useful to find tangent planes to surfaces and tangent lines.

Physical Interpretation of Directional Derivative

In the figure below, ∂f/ ∂x represents the slope of the “surface” f(x,y) in

the x-direction at point P, ∂f/ ∂y represents the slope of the “surface” f(x,y) in

the y-direction at point P, and ∂f/ ∂z represents the slope of the “surface” f(x,y) in

the z-direction at point P.

If u is any arbitrary unit vector, then the directional derivative of the function in the

direction of u , D_u f(x,y), represents the slope of the function in the direction of u

and can be calculated by the dot product , grad F · u = n · u .

Click here for examples.

Return to Calculus 3 Notes

Copyright © 2017 Richard C. Coddington

All rights reserved.