先寫一下 notations
$f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ 則 Jacobian matrix 為
$$ \mathbf{D}f(x)=\left[ \begin{array}{ccc} \frac{\partial f_1}{\partial x_1} && ... && \frac{\partial f_1}{\partial x_n} \\ \vdots && \ddots && \vdots \\ \frac{\partial f_n}{\partial x_1} && ... && \frac{\partial f_n}{\partial x_n} \\ \end{array} \right] $$
而 “determinant 的 Jacobian matrix“ 稱 Jacobian 寫為 $J_f(x)$:
$$ J_f(x):=\det(\mathbf{Df(x)}) \\ \text{or}\quad := \frac{\partial(f_1,...,f_n)}{\partial(x_1,..,x_n)} $$
Let $A\subset\mathbb{R}^n$ be an open bounded set with volume and let $g:A\rightarrow\mathbb{R}^n$ be a $\mathcal{C}^1$ mapping that is one-to-one. Assume that $J_g(x)\neq0$ for all $x\in A$ and that $|J_g(x)|$ and $1/|J_g(x)|$ are bounded on $A$. Let $B=g(A)$ and assume that $B$ has volume. If $f:B\rightarrow\mathbb{R}$ is bounded and integrable, then $(f\circ g)|J_g|$ is integrable on $A$ and
$$ \int_B f=\int_A(f\circ g)|J_g| $$
, that is,
$$ \int_B f(y_1,...,y_n)dy_1...dy_n=\int_Af(g(x_1,...,x_n)) \left|\frac{\partial(g_1,...,g_n)}{\partial (x_1,...,x_n)}\right| dx_1...dx_n $$
⚠️ 課本在 RHS 的 Jacobian 應該少了絕對值, 或參考下圖 [來源]
記住:
$$ dxdy=\left|\frac{\partial(x,y)}{\partial(u,v)}\right|dudv=|J_g|dudv \\ ,\quad\text{where }(x,y)=g(u,v) $$
注意到 Jacobian matrix $\mathbf{D}g$ 物理意義為單位 $(u,v)$ 的變化造成的 $(x,y)$ 變化
而絕對值的 determinant 一個 square matrix, i.e. $|\det(A)|$, 物理意義為 $A$ 所形成的平行六面體之體積
所以 $|J_g|=|\det\mathbf{D}g|$ 可以理解為在 $(u,v)$ 上單位體積的變化造成的 $(x,y)$ 體積的變化