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Home Vector Calculus Directional Derivatives and Gradients Theorem 1: Directional Derivatives (Two Variables)
 
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Theorem 1: Directional Derivatives (Two Variables)

THEOREM 1

Suppose z = f(x,y) is smooth at (a, b). Then for any unit vector

U = cos αi + sin αj,

the directional derivative fU(a, b) exists and

13_vector_calculus-8.gif

PROOF

Let

U = cos αi + sin αj.

Write x, y, and z as functions of f,

x = α + t cos α,

y = a + t sin α,

z = f(a + r cos α,b + t sin α).

Then by the Chain Rule,

13_vector_calculus-10.gif
Home Vector Calculus Directional Derivatives and Gradients Theorem 1: Directional Derivatives (Two Variables)

Last Update: 2006-11-25