question regarding total derivates

**tombrownington** · January 9th 2009, 10:16 PM

Suppose I have variables x, y and z given parametrically as functions of two variables u and v.

How can I get the total derivative of z with respect to the variable x, without having to find u and v in function of x and y?

I know total derivatives can be applied when the different variables are functions of 1 parameter. Here there are two parameters, u and v. How do I proceed?

Thanks in advance.

**Danny** · January 10th 2009, 08:09 AM

Originally Posted by tombrownington

Suppose I have variables x, y and z given parametrically as functions of two variables u and v.

How can I get the total derivative of z with respect to the variable x, without having to find u and v in function of x and y?

I know total derivatives can be applied when the different variables are functions of 1 parameter. Here there are two parameters, u and v. How do I proceed?

Thanks in advance.

If $x, y, \;\text{and}\; z$ are given as functions of $u\; \text{and}\; v$ it is often the case that you can't eliminate $u\; \text{and}\; v$ . To calculate the derivative of

$\frac{\partial z}{\partial x}$

you use Jacobians

$\frac{\partial z}{\partial x} = \frac{\partial (z,y)}{\partial (x,y)} = \frac{\partial (z,y)}{\partial (u,v)} \cdot \frac{\partial (u,v)}{\partial (x,y)} = \frac{\partial (z,y)}{\partial (u,v)} / \frac{\partial (x,y)}{\partial (u,v)}$

where the Jacobian is defined as

$\frac{\partial (x,y)}{\partial (u,v)} = \begin{array}{| c c |}x_u & x_v\\y_u&y_v\end{array}$

where subscripts are partial differentiation.

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