Differentiating expressions involving multivariable vector valued functions

[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!

 

[HallsofIvy]

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.

 

[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!