Partial derivatives transformation

nmsurobert · Jan 19, 2015

Homework Statement

Handwritten worksheet solution: partial derivatives of t and x with respect to t and x expressed in terms of the primed partial derivatives, using chain rule terms dt'/dt, dx'/dt, dt'/dx and dx'/dx.

Homework Equations

included in the first picture

The Attempt at a Solution

Handwritten derivation on a whiteboard: d/dt = (dt'/dt) d/dt' + (dx'/dt) d/dx' = gamma d/dt' + gamma v d/dx'; d/dx = (dt'/dx) d/dt' + (dx'/dx) d/dx' = -(gamma v / c^2) d/dt' + gamma d/dx'; below a line, Solution: (gamma d/dt' + gamma v d/dx')^2 - c^2 (-(gamma v / c^2) d/dt' + gamma d/dx')^2.

i feel confident in my answer to part "a". i pretty much just did what the u and v example at the top of the page did. but for part "b" i tried to distribute and collect like terms and what not and i ended up with a huge mess. so i can anyone tell me if I am on the right track. and that the end product should like look a huge mess of terms.

thanks!

RUber · Jan 20, 2015

##\frac{\partial^2}{\partial t^2} \neq \left(\frac{\partial}{\partial t}\right)^2##
You have to apply the derivative twice.

nmsurobert · Jan 23, 2015

so i figured out a solution and i thought i would add it here in case someone googling for information stumbles upon this thread.

Handwritten derivation on a whiteboard: the wave operator d²/dt² minus c² d²/dx² rewritten in primed coordinates with gamma and velocity v, simplified step by step to d²/dt'² minus c² d²/dx'².

i get from the second step to the third step (where i am left only with derivatives and a c^2) using the definition of gamma given to me and some rigorous algebra.
took me a solid two days to figure this out.

Partial derivatives transformation

Homework Statement

Homework Equations

The Attempt at a Solution

Discussion

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