in vector calculus,when variable changes by function $g : S \rightarrow U$ $\int_C \textbf{F}(\textbf{s})\cdot \textbf{ds} = \det{J\left( g \right)} \int_{g(C)} \textbf{F}(\textbf{u})\cdot \textbf{du} $
Then if for complex analysis, see complex as real vector the real linear transform $ T : Z \rightarrow W\space(T: \mathbb{R^2} \rightarrow \mathbb{R^2}$) could T work like as under equations? $ \oint f(z) dz = \det{T} \oint f(w)dw $