I have recently encountered a problem in complex analysis. Suppose we have domains $D$ and $\tilde{D},$ and let $\phi: D \to \tilde{D}$ be a conformal mapping. Suppose we also have functions $f,g\in C^{\infty}(D),$ then the question is to prove the following identity holds: $$\int_{D}\bigtriangledown f(z)\cdot \bigtriangledown g(z) dz= \int_{\tilde{D}}\bigtriangledown f \circ \phi ^{-1}(z)\cdot \bigtriangledown g \circ \phi^{-1}(z)dz.$$
My attempts:
I was trying to perform a change of variable to prove this identity. If I let $t=\phi^{-1}(z),$ then the RHS of this identity will become $$\int_{\tilde{D}}\bigtriangledown f \circ \phi ^{-1}(z)\cdot \bigtriangledown g \circ \phi^{-1}(z)dz=\int_{{D}}\bigtriangledown f(t)\cdot \bigtriangledown g(t) \frac{dz}{dt}dt,$$
but I'm not quite sure how to deal with the Jacobian term $\frac{dz}{dt}$. Could anybody give me some hints or solutions about how to prove this identity? Thank you very much for your help in advance!