# Directional derivative of vector valued functions?

Just to confirm, is the directional derivative of a vector valued function calculated as Lu ? where L is the Frechet derivative , and u is the unit vector in the direction.

There seem to be a lot of sources for a real valued function's directional derivative, but very little on vector valued function in Rm.

it says DuF(directional derivative) = lim (F(x + tu) - F(x)) / t as t->0
if F is diff'ble, then -] L , then just set h = tu, we get the desired result?..

HallsofIvy
Homework Helper
There's a reason why you find "very little on vector valued function in Rm"!

If f is a differentiable function from Rm to Rn, then the derivative of f is a linear function from Rm to Rn which can be represented by a n by m matrix. In particular, the directional derivative of a n dimensional vector valued function on Rm in the direction of vector m dimensional vector v, is an n dimensional vector given by the product of the matrix representing the derivative and the vector v, represented as a column matrix.

so that means, if the Fretchet derivative is L, and the directional vector is u, it is just Lu.

Sorry for bringing this thread back from the dead, but I would like to clarify a doubt, if possible.

In order to calculate the directional derivative of a scalar field, we must use a unit vector to define the direction.

What happens in the case of a vector field? Does the direction also have to be defined by a unit vector, or we can use any vector?

If anyone could help, I would appreciate.

Consider the derivative operator $$D_v$$. We require it to be linear in the direction of integration, v. So $$D_{2v} = 2 D_v$$ and $$D_{u+v} = D_u + D_v$$. In fact, this correspondence is taken to be an isomorphism in differential geometry, where tangent vectors are defined to be directional derivative operators.

I'll use 2 examples:

Scalar field

f(x,y)=x$$^{3}$$y$$^{2}$$

To find the directional derivative for this function in the point P(-1,2), in the direction of the vector a=4i-3j, we must start by finding a unit vector, u, with the same direction as a.

u=$$\frac{1}{\left\right\|a\|}$$a = $$\frac{1}{5}$$ (4i-3j) = $$\frac{4}{5}$$i - $$\frac{3}{5}$$j

The result is:

Du f(x,y) = 3x$$^{2}$$y$$^{2}$$ ($$\frac{4}{5}$$) + 2x$$^{3}$$y ($$\frac{-3}{5}$$)

For P(-1,2)

Du f(-1,2) = 12

Vector field

f(x,y)= (y logx, x$$^{3}$$ - 3y)

Suppose we want to find the directional derivative for this function in the point P(1,2), in the direction of the vector v=1i+4j.

My doubt is: do we need to find a unit vector or we simply use v=(1,4).

I solved it like this (but I'm not sure it's right):

Dv f(x,y) = [[y/x, logx][3x$$^{2}$$, -3]]*[[1][4]]
Dv f(1,2) = [[2][-9]]

Sorry for the matrix notation, but I couldn't get it right with LaTeX.

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