Let X be an open set in R n , F: X ⊆ Rn → Rn a vector field on X, and a ∈ X. If v is any unit vector in Rn, we define the directional derivative of F at a in the direction of v, denoted DvF(a), by provided that the limit exists. Exercises 31–34 involve directional derivatives of vector fields.
Show that the directional derivative of a vector field F is the vector whose components are the directional derivatives of the component functions F1, . . . , Fn of F, that is, that DvF(a) = (DvF1(a), DvF2(a), . . . , DvFn(a)).
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