4. Introduction to multi-variable calculus#
4.1. Function#
4.2. Limit#
4.3. Derivatives#
4.4. Integrals#
4.5. Theorems#
4.5.1. Green’s lemma#
\[\begin{split}\begin{aligned}
\int_{S} \frac{\partial F}{\partial y} dx dy & = - \oint_{\partial S} F dx \\
\int_{S} \frac{\partial G}{\partial x} dx dy & = \quad \oint_{\partial S} G dy
\end{aligned}\end{split}\]
Proof for simple domains.
In a simple domain in \(x\), so that the closed contour \(\partial S\) is delimited by the curves \(y=Y_1(x)\), \(y=Y_2(x) > Y_1(x)\), for \(x \in [x_1, x_2]\),
\[\begin{split}\begin{aligned}
\int_{S} \frac{\partial F}{\partial y} dx dy
& = \int_{x=x_1}^{x_2} \int_{y = Y_1(x)}^{Y_2(x)} \frac{\partial F}{\partial y} dy \, dx = \\
& = \int_{x=x_1}^{x_2} \left[ F(x,Y_2(x)) - F(x,Y_1(x)) \right] dx = \\
& = - \int_{x=x_1}^{x_2} F(x,Y_1(x)) - \int_{x=x_2}^{x_1} F(x, Y_2(x)) dx = \\
& = - \oint_{\partial S} F(x,y) dx
\end{aligned}\end{split}\]
Proof for generic domain
todo
In a simple domain in \(y\), so that the closed contour \(\partial S\) is delimited by the curves \(x=X_1(y)\), \(x=X_2(y) > X_1(y)\) for \(y \in [y_1, y_2]\),
\[\begin{split}\begin{aligned}
\int_{S} \frac{\partial G}{\partial x} dx dy
& = \int_{y=y_1}^{y_2} \int_{x = X_1(y)}^{X_2(y)} \frac{\partial G}{\partial x} dx \, dy = \\
& = \int_{y=y_1}^{y_2} \left[ G(X_2(y),y) - G(X_1(y),y) \right] dy = \\
& = \int_{y=y_1}^{y_2} G(X_1(y),y) dy + \int_{y=y_2}^{y_1} G(X_2(y),y) dy = \\
& = \oint_{\partial S} G(x,y) dy
\end{aligned}\end{split}\]