# D'alembert's solution to the wave equation, need help on Chain Rule

## Homework Statement

Please have a look at the picture attach, which shows the proof of the D'alembert's solution to the wave equation. If you can't open the open,
https://www.physicsforums.com/attachment.php?attachmentid=54937&stc=1&d=1358917223

click onto this link:http://www-solar.mcs.st-and.ac.uk/~alan/MT2003/PDE/node12.html

The part I have problem is the taking the second paritial derivative of u with respect to x.
Ux=du/dz+du/dn, does this mean that Ux is a function of z and n, that is, Ux(z,n), and z(x,t) and n(x,t)?

so later when I taken the second derivative, then I apply chain rule?

Thank you

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## Answers and Replies

tiny-tim
Science Advisor
Homework Helper
hi kougou! (try using the Quick Symbols box next to the Reply box )
The part I have problem is the taking the second paritial derivative of u with respect to x.
Ux=du/dz+du/dn, does this mean that Ux is a function of z and n, that is, Ux(z,n), and z(x,t) and n(x,t)?

so later when I taken the second derivative, then I apply chain rule?

i'm not really following your question ∂u/∂x can be considered either a function of x and y, or a function of ζ and η

since you want a final result in terms of ζ and η, you'll have to convert the derivatives wrt x and y into derivatives wrt ζ and η (and yes, you use the chain rule)