Having problems with the (I think) general chain rule

[richyw]

Homework Statement

Consider the transformation $\mathbf{x}=G(\mathbf{u}), \text{ where } \mathbf{x}=(x_1,x_2,x_3),\:\mathbf{u}=(u_1,u_2,u_3)$ given by$$x_1=u_1+u_3^2$$$$x_2=u_3-u_1^2$$$$x_3=u_1+u_2+u_3$$

I need to compute the derivative of this transformation, and then show that the transformation is locally invertible if $u_1u_3>0$

Homework Equations

unsure really.

The Attempt at a Solution

well x is a function of $x_1,x_2,x_3$ and all the $x_j's$ are functions of $u_1,u_2,u_3$. So I am dealing with a map $\mathbb{R}^3$ to $\mathbb{R}^3$ right?

sorry I'm really lost on what this question is asking. mostly focusing on the firsrt part right now (the derivative)

 

[LCKurtz]

Homework Statement

Consider the transformation $\mathbf{x}=G(\mathbf{u}), \text{ where } \mathbf{x}=(x_1,x_2,x_3),\:\mathbf{u}=(u_1,u_2,u_3)$ given by$$x_1=u_1+u_3^2$$$$x_2=u_3-u_1^2$$$$u_1+u_2+u_3$$

I need to compute the derivative of this transformation, and then show that the transformation is locally invertible if $u_1u_3>0$

Homework Equations

unsure really.

The Attempt at a Solution

well x is a function of $x_1,x_2,x_3$ and all the $x_j's$ are functions of $u_1,u_2,u_3$. So I am dealing with a map $\mathbb{R}^3$ to $\mathbb{R}^3$ right?

sorry I'm really lost on what this question is asking. mostly focusing on the firsrt part right now (the derivative)

I suppose that third line is supposed to read $x_3=u_1+u_2+u_3$. Is that what is causing your problems? Can you calculate the Jacobian now?

 

[richyw]

that's a typo yes, but not what is causing my problem.

 

[richyw]

$$(x_1,x_2,x_3)=G(u_1,u_2,u_3)$$
$$D\mathbf{x}=\left[\begin{matrix} \frac{\partial x_1}{\partial u_1} & \frac{\partial x_1}{\partial u_2} & \frac{\partial x_1}{\partial u_3} \\ \frac{\partial x_2}{\partial u_1} & \frac{\partial x_2}{\partial u_2} & \frac{\partial x_2}{\partial x_3} \\ \frac{\partial x_3}{\partial u_1} & \frac{\partial x_3}{\partial u_2} & \frac{\partial x_3}{\partial u_3}\end{matrix}\right]$$

 

[richyw]

is that the jacobian?

 

[LCKurtz]

$$(x_1,x_2,x_3)=G(u_1,u_2,u_3)$$
$$D\mathbf{x}=\left[\begin{matrix} \frac{\partial x_1}{\partial u_1} & \frac{\partial x_1}{\partial u_2} & \frac{\partial x_1}{\partial u_3} \\ \frac{\partial x_2}{\partial u_1} & \frac{\partial x_2}{\partial u_2} & \frac{\partial x_2}{\partial x_3} \\ \frac{\partial x_3}{\partial u_1} & \frac{\partial x_3}{\partial u_2} & \frac{\partial x_3}{\partial u_3}\end{matrix}\right]$$

is that the jacobian?

Yes. You can read about the Jacobian and its properties here:

http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant

 

[richyw]

ok cool, thanks a lot!

I'm sure i'll be able to hack through the rest of it. I'm quite frustrated with my textbook right now (folland). It seems to be more of a reference text than something that I can actually learn from. Still looking for a companion text, or even better something like khan academy.