Not really a homework problem... just a general question (this seemed like the place to put it...). Say I have three functions:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]f,g,h:\mathbb{R}^2\rightarrow\mathbb{R}^3[/tex]

and an expression along the lines of:

[tex]\left\langle f(u_1,u_2),g(u_1,u_2)\right\rangle h(u_1,u_2)[/tex]

What differentiation rules allow me to compute

[tex]\frac{\partial}{\partial \vec{u}}(\left\langle f(u_1,u_2),g(u_1,u_2)\right\rangle h(u_1,u_2))[/tex]

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:

[tex]\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}}[/tex]

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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Differentiating expressions involving multivariable vector valued functions

**Physics Forums | Science Articles, Homework Help, Discussion**