Thread: Simple PDE Question

1. Simple PDE Question

I ran across the following in a book, but I don't quite understand why we have fx+p fz=0 (and similarly with y). A little guidance would be greatly appreciated.

2. Re: Simple PDE Question

Originally Posted by Hal2001
I ran across the following in a book, but I don't quite understand why we have fx+p fz=0 (and similarly with y). A little guidance would be greatly appreciated.
The assumptions is that z is a function of both x and y, that is

$\displaystyle z(x,y) \implies f(x,y,z(x,y),a,b)=0$

Now just use the chain rule

$\displaystyle \frac{\partial }{\partial }f(x,y,z(x,y),a,b)= \frac{\partial f}{\partial x}\frac{\partial x}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial x}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial x}+\frac{\partial f}{\partial a}\frac{\partial a}{\partial x}+\frac{\partial f}{\partial b}\frac{\partial b}{\partial x}=0$

This reduces to

$\displaystyle \frac{\partial }{\partial x}f(x,y,z(x,y),a,b)= \frac{\partial f}{\partial x}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial x} =0$

3. Re: Simple PDE Question

got it thanks. Didn't realize at the time that they were defining z as a function of x and y. But it makes sense now.