Differentiating expressions involving multivariable vector valued functions

1. Oct 16, 2008

flyingtabmow

Not really a homework problem... just a general question (this seemed like the place to put it...). Say I have three functions:

$$f,g,h:\mathbb{R}^2\rightarrow\mathbb{R}^3$$

and an expression along the lines of:

$$\left\langle f(u_1,u_2),g(u_1,u_2)\right\rangle h(u_1,u_2)$$

What differentiation rules allow me to compute

$$\frac{\partial}{\partial \vec{u}}(\left\langle f(u_1,u_2),g(u_1,u_2)\right\rangle h(u_1,u_2))$$

My problem is that I'm unclear on how to order/transpose the Jacobians and vectors such that all of the multiplications make sense. It's possible to order them as follows:

$$\left\langle f(\vec{u}),g(\vec{u})\right\rangle \frac{\partial h}{\partial \vec{u}} + h(\vec{u})\cdot f(\vec{u})^\mathrm{T}\cdot \frac{\partial g}{\partial \vec{u}} + h(\vec{u})\cdot g(\vec{u})^\mathrm{T}\cdot \frac{\partial f}{\partial \vec{u}}$$

Everything works out there, and the result is a 3x2 matrix. However I'm not clear on what rules allow me to actually arrive at this result (other than moving things around until it all fits).

If anyone knows a good (preferably online) resource or book that discusses these sorts of expressions, that would be very helpful as well.

Thanks!

2. Oct 16, 2008

HallsofIvy

Staff Emeritus
No rules. You have to go back to the definition. Your function is from R2 to R3 so the derivative, at a given point in R2 is the linear transformation from R2 to R3 that most closely approximates f- if f is differentiable so that is unique.

In a particular coordinate system, so that $F(u_1,u_2)= \left< f(u_1,u_2), g(u_1,u_2), h(u_1,u_2)\right>$ the derivative, at $(u_1, u_2)$ can be represented by the 2 by 3 matrix
$$\left[\begin{array}{cc}\frac{\partial f}{\partial u_1} & \frac{\partial f}{\partial u_2} \\ \frac{\partial g}{\partial u_1} & \frac{\partial g}{\partial u_2} \\ \frac{\partial h}{\partial u_1} & \frac{\partial h}{\partial u_2}\end{array}\right]$$
Strictly speaking, the derivative is the linear transformation represented by that matrix.

Again, that is at a particular point in R2- at a particular (u1, u2). If you want the derivative FUNCTION, you will have to think of it as a function from R2 to the set of matrices from R2 to R3 which can, itself, be represented as a function from R2 to R6.

3. Oct 16, 2008

flyingtabmow

Sorry, I must not have been clear. The three functions $$f$$, $$g$$, and $$h$$ are each from $$\mathbb{R}^2\rightarrow\mathbb{R}^3$$. That is, they each have coordinate functions $$f_i$$, $$g_i$$, and $$h_i$$ ($$f$$, $$g$$, and $$h$$ are not the coordinate functions of some $$F:\mathbb{R}^2\rightarrow\mathbb{R}^3$$). The use of the angle bracket notation was to signify an inner product, not different components of a vector. Thus $$\left\langle f(u_1,u_2),g(u_1,u_2)\right\rangle$$ is a scalar, and $$\left\langle f(u_1,u_2),g(u_1,u_2)\right\rangle h(u_1,u_2)$$ is a column vector, and I'm wondering how to compute

$$\frac{\partial}{\partial \vec{u}}(\left\langle f(u_1,u_2),g(u_1,u_2)\right\rangle h(u_1,u_2))$$

and what differentiation rules apply.

Thanks!