Hi, I’m a bit confused.(adsbygoogle = window.adsbygoogle || []).push({});

I am familiar with the chain rule: if y=f(g(t,x),h(t,x)) then dy/dt=dy/dg*dg/dt+dy/dh*dh/dt

To show that an equation is invariant under a galiliean transform, it’s partially necessary to show that the equation takes the same form both for x and for x’=x-v(T). So if you have a wave equation for E which applies for x, and t, you want to show that the wave equation, with all of its first and second derivatives also applies for x’ and t’.

For example if you look at question 16 b in the following link, they ask to show that the wave equation is not invariant under Galilean transforms. What I don’t understand is in this question why are they taking the derivative of E with respect to x and t, rather than with respect to x’ and t’. We already know the wave equation takes the correct form for x and t. We want to show that it doesn’t take the correct form for x’ and t’, so then why start off taking the derivative with respect to x and t, and muck about using the chain rule rather than taking the derivative with respect to x’, and t’ (which is what you’re really interested in).

http://stuff.mit.edu/afs/athena/course/8/8.20/www/sols/sol1.pdf

I just don’t get why they take the derivative with respect to x and t, when that’s not what you’re really interested in, you’re interested in the form of the derivatives with respect to x’ and t’. Where does the idea to use the chain rule in this case comes from at all? All of the solutions for these types of problems seem to use the chain rule, I don’t get where the natural impetus to take the derivative in the wave equation with respect to variables you already know will work in the wave equations, rather than the ones which have undergone the Galilean transform.

It’s been a very long time since I’ve done this type of math/physics. Thanks.

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# Use of chain rule in showing invariance.

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