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.
Let F = x i + y j + z k. Show that DvF(a) = v for any point a ∈ R3 and any unit vector v ∈ R3. More generally, if F = (x1, x2, . . . , xn), a = (a1, a2, . . . , an), and v = (v1, v2, . . . , vn), show that DvF(a) = v.
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