the multivariate change-of-variables formula
Often you transform several random variables at once: from rectangular coordinates (X, Y) to polar (R, Theta), or from a pair of measurements to their sum and difference. The multivariate change-of-variables formula gives the joint density of the new variables in one go. It is the higher-dimensional sibling of the single-variable Jacobian rule.
The logic is identical: probability is volume, and a transformation stretches or squeezes little boxes of volume. In one dimension the stretch factor was |dx/dy|; in n dimensions it is the absolute value of the determinant of the Jacobian matrix — the matrix of all partial derivatives of the old variables with respect to the new ones. That determinant measures how much a tiny n-dimensional box of volume gets scaled by the map. So if (U, V) = T(X, Y) is a smooth one-to-one map with inverse (X, Y) = T^{-1}(U, V), then f_{U,V}(u, v) = f_{X,Y}(x, y) times |det J|, with everything evaluated at the inverse image and J the Jacobian of the inverse.
This formula is the engine behind some of the most useful results in probability: it is how you derive that a ratio of independent normals is Cauchy, how the polar trick (Box-Muller) turns two uniforms into two normals, and how sums and differences are tackled jointly before marginalising. Whenever you need the joint law of several functions of several variables, this is the tool.
From (X, Y) to (U, V) = (X + Y, X - Y): the inverse is X = (U + V)/2, Y = (U - V)/2. The Jacobian matrix of the inverse has determinant -1/2, so |det J| = 1/2 and f_{U,V}(u, v) = f_{X,Y}((u+v)/2, (u-v)/2) times 1/2. The 1/2 accounts for how the rotate-and-scale map changes area.
The absolute determinant of the Jacobian is the multi-dimensional volume-stretch factor.
The map must be one-to-one on the region where the density lives, and you must use the determinant of the FULL Jacobian, not partials separately. To get just one new variable, transform to a convenient pair and then marginalise out the spare.