Suppose T is a $1-1$ ($\mathcal{C'}$ mapping of an open set $E \subset R^k$ into $R^k$ such that $J_{T(x)} \ne 0$ for all $x \in E$. If $f$ is a continuous function on $R^k$ whose support is compact and lies in $T(E)$, then
$\int_{R^k}f(y)dy = \int_{R^k}f(T(x))|J_T(x)|dx$
This is the generalization of the change of variables in the one-dimensional case.
In one dimensional case we have $\int_{R^1}f(y) dy = \int_{R^1}f(T(x))T'(x) dx$,
So if we try to generalize this then $T'(x)$ becomes a $k \times k$ matrix .What I don't understand is that why are we taking the determinant of the function $T'(x)$?Is it solely because $T'(x)$ is a $k \times k$ matrix which cannot be multiplies with $f(T(y))$ to get a scalar or is there an intuition behind this?