How Does Changing Variables Affect Integrals in Calculus?

[LCSphysicist]

Homework Statement

I am having a little trouble to see how the changes of variables works here.

Relevant Equations

Functionals.

Be $x = x(u,v) y = y(u,v)$, if $F = \int f(x,y,y')dx$ and the Jacobian's determinant different of zero, $v = v(u)$
${\Large {J = \int F[x,y,y']dx ---> \int F[x(u,v),y(u,v),\frac{y_{u} + y_{v}v'}{x_{u} + x_{v}v'}](x_{u} + x_{v}v')du}}$

The last term in the bracket is confusing me, how to get it?

 

[PeroK]

You have $x = x(u, v(u))$ and you need $\frac{dx}{du}$. Can you do that differentiation?

 

[LCSphysicist]

You have $x = x(u, v(u))$ and you need $\frac{dx}{du}$. Can you do that differentiation?

Oh, so we can do y' = dy/dx = dy/du/dx/du, i was confusing terms, but now it is ok, thx