# How jacobian transformation have been used in this problem

Let $t$ and $a$ be positive real numbers. Define $B_a = \{x=(x_1,x_2,x_3,\dots,x_n) \in\mathbb R^n\mid x_1^2 + x_2^2+\dots+x_n^2 \leq a^2\}$. Then for any compactly supported continous function $f$ on $\mathbb R^n$ Then the following is correct $\int_{ B_a} {f(xt) dx} =\int_{B_{ta}} {f(x) t^{-n} dx}$

But I can't understand.. I have seen its solution it has written directly Jacobian transformation $= t^{-n}$ How can we get $t^{-n}$ in RHS. Plz tell me how the Jacobian has been used

The determinant of the (Jacobian of the) linear map $T(x)=tx$ ($x\in\Bbb R^n$) is $t^n$. If you rewrote your equation as $$\int_{B_{ta}} f(x)\,dx = \int_{B_a} f(tx)t^n\,dx$$ it would fit the change-of-variables formula precisely (assuming $t>0$, of course).