Let X be an open set in R n , F: X ⊆ Rn → Rn a vector field on...

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 D_vF(a), by provided that the limit exists. Exercises 31–34 involve directional derivatives of vector fields.

(a) In analogy with the directional derivative of a scalar-valued function defined in §2.6, show that

(b) Use the result of part (a) and the chain rule to show that, if F is differentiable at a, then

D _v F(a) = DF(a)v,

where v is interpreted to be an n × 1 matrix. (Note that this result makes it straightforward to calculate directional derivatives of vector fields.)