Seem to remember the following equation held:
$f(u) = {dx\over du} f(x)$
if one is give the probability distribution of x and a relationship between x and u the pdf of u can be derived. Sorry can't remember if it has a name. Does this extend to higher dimensions? i.e let u be a known function(x,y,z) and x, y and z have know probability distribution. Is the pdf of u then simply:
$f(u) = {dx\over du} f(x) + {dy\over du} f(y) +{dx\over du} f(x) $
If not how does on obtain the pdf of u? Thanks.
Your guess for the pdf is incorrect. If you have an invertible transformation $ (u,v,w)=\Phi(x,y,z)$, then the joint pdf of $(u,v,w)$ is $$ \left|\det \frac{\partial (x,y,z)}{\partial (u,v,w)}\right| (f\circ \Phi^{-1})$$ where the first factor involves the Jacobian determinant. This is known as the change of variables formula for integration.
If instead you have a transformation $u=\Phi(x,y,z)$, then you have to integrate $|\nabla \Phi|^{-1} f(x,y,z)$ over the level set $\{\Phi(x,y,z)=c\}$ to obtain the density at $u=c$. This can be quite difficult in practice. See: coarea formula.