The name directional suggests they are vector functions. However, since a directional derivative is the dot product of the gradient and a vector it has to be a scalar. But, in my textbook, I see the special case of the directional derivatives F x ( x , y , z ) and F y ( x , y , z ) being treated as vectors. I want a clarification for this.

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The name directional suggests they are vector functions. However, since a directional derivative is the dot product of the gradient and a vector it has to be a scalar. But, in my textbook, I see the special case of the directional derivatives       F    x    (  x  ,  y  ,  z  ) and       F    y    (  x  ,  y  ,  z  ) being treated as vectors. I want a clarification for this.

fobiosofia3ql · Accepted Answer

Typically, we think of the directional derivative of a scalar-valued function   f  :            R        n    →      R   in a (unit) direction   v  ∈            R        n   at a point   x  ∈            R        n  ; this is just defined as  ∇  f            |        x    ⋅  vAt a particular point x, this is just a scalar; we can also view   ∇  f  ⋅  v more generally as another function             R        n    →      R  , assigning to each point x f’s derivative in direction v at that point.In the case that f is a vector-valued function   f  :            R        n    →            R        m  , we could define a sort of “vector directional derivative” (nonstandard vocabulary)             R        n    →            R        m   of f at x in direction v by taking the directional derivative of each component function       f    i    :            R        n    →      R   and assembling these into a vector in             R        m  . Equivalently, we could take the   m  ×  n Jacobian matrix J(f) of f’s partial derivatives, whose (i,j)-th entry is             ∂              f        i                    ∂              x        j            ; then this “vector directional derivative” is simply equal to   J  (  f  )    v, as a matrix product.

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