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Math Help - Simple PDE Question

  1. #1
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    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.
    Simple PDE Question-misc.png
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  2. #2
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    Re: Simple PDE Question

    Quote Originally Posted by Hal2001 View Post
    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.
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    The assumptions is that z is a function of both x and y, that is

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

    Now just use the chain rule

    \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

    \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
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  3. #3
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    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.
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