Green's theorem for rectangles and Cauchy's theorem on a rectangle

Let \[ R=[a,b]\times[c,d] \] and let \(P,Q\) have continuous first partial derivatives on a neighborhood of \(R\). Green’s theorem for the positively oriented boundary of the rectangle is \[ \boxed{ \int_{\partial R}P\,dx+Q\,dy =\iint_R(Q_x-P_y)\,dA.} \]

For a rectangle this follows directly from the one-variable fundamental theorem of calculus. Indeed, summing the integrals on the horizontal sides gives \[ \int_a^b P(x,c)\,dx-\int_a^b P(x,d)\,dx =-\int_a^b\int_c^d P_y(x,y)\,dy\,dx, \] while summing the integrals on the vertical sides gives \[ \int_c^d Q(b,y)\,dy-\int_c^d Q(a,y)\,dy =\int_c^d\int_a^b Q_x(x,y)\,dx\,dy. \] Adding and using Fubini yields the displayed formula.

Now let \(f=u+iv\) be holomorphic on a neighborhood of \(R\). Then \[ f(z)\,dz=(u+iv)(dx+i\,dy) =(u\,dx-v\,dy)+i(v\,dx+u\,dy). \] Applying Green’s theorem to the real part gives \[ \int_{\partial R}(u\,dx-v\,dy) =\iint_R(-v_x-u_y)\,dA=0, \] because the Cauchy–Riemann equation \(v_x=-u_y\) holds. For the imaginary part, \[ \int_{\partial R}(v\,dx+u\,dy) =\iint_R(u_x-v_y)\,dA=0 \] by \(u_x=v_y\). Therefore \[ \boxed{\int_{\partial R}f(z)\,dz=0,} \] which is Cauchy’s theorem for a rectangle.