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. |
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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.
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Copyright © 2017 Richard C. Coddington
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