I remember I learned how to do the inverse of this, but I'm looking two things, one is the right way for the next change of variable, second the demonstration.
Z function give the third axis in function of x and y
$$Z(x, y)$$
We travel the space so we have two T (travel) functions who tell the current x/y coords:
$$x = T_x(t)$$ $$y = T_y(t)$$
The integral on time:
$$\int_{t_1}^{t_2} f(T_x, T_y, Z(T_x, T_y)) \,dt$$
I want to change the variable from dt to dx and dy.
From what I remember, because I didn't learn very well this years ago, if one variable can be written in other we have this:
$$dt(x, y)=\frac{\partial dt}{\partial dx}dx+\frac{\partial dy}{\partial dy}dx$$
Which could lead to:
$$\int_{x_1}^{x_2} \frac{f(T_x, T_y, Z(T_x, T_y))}{\frac{\partial dx}{\partial dt}}dx+\int_{y_1}^{y_2} \frac{f(T_x, T_y, Z(T_x, T_y))}{\frac{\partial dy}{\partial dt}}dy$$
But, even if that is right, I would not have idea why....
If someone ask when this is useful, this comes from other system, and the speed is in function of Z, that means if you have $dx/dt$ is something that can be replaced with a function that does not have time as a variable and you only need to integrate on x/y.
Thx!