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Vector Calculus: An Introduction

Daniel Daners

Section 11.1 The Divergence Theorem in the Plane

Suppose that \(\vect f=(f_1,f_2)\) is a smooth vector field. Applying Green’s theorem to the field \((-f_2,f_1)\) we obtain
\begin{equation*} \iint_D\Bigl(\frac{\partial}{\partial x_1}f_1 +\frac{\partial}{\partial x_2}f_2\Bigr)\,dx_1\,dx_2 =\int_{\partial D}f_1\,dx_2-f_2\,dx_1\text{.} \end{equation*}
We want to get an interpretation of the integral on the right hand side. To do so suppose that
\begin{equation*} \vect\gamma(t)=(\gamma_1(t),\gamma_2(t)),\qquad t\in[a,b] \end{equation*}
is a regular parametrisation of some part \(C\subset\partial D\) consistent with the orientation of \(\partial D\text{.}\) By definition of the line integral we have
\begin{equation*} \int_{C}f_1\,dx_2-f_2\,dx_1 =\int_a^bf_1(\gamma_1(t),\gamma_2(t))\gamma_2'(t) -f_2((\gamma_1(t),\gamma_2(t))\gamma_1'(t)\,dt\text{.} \end{equation*}
We can view the integrand on the right hand side as a dot product of \(\vect f\) with the vector \((\gamma_2'(t),-\gamma_1'(t))\text{.}\) We rescale the latter to unit length and define
\begin{equation*} \vect n(t):=\frac{1}{\|\vect\gamma'(t)\|} \begin{bmatrix} \gamma_2'(t) \\-\gamma_1'(t) \end{bmatrix}\text{.} \end{equation*}
Observe that \(\vect n(t)\) is perpendicular to the positive unit tangent vector \(\vect\tau(t)=\vect\gamma'(t)/\|\gamma'(t)\|\text{.}\) As it is outward pointing, it is called the outward pointing unit normal to \(D\text{.}\) Both vectors are shown in Figure 11.1.
Figure 11.1. Positive tangent and outward pointing unit normal vectors.
Hence we can rewrite our integral as
\begin{equation*} \int_{C}f_1\,dx_2-f_2\,dx_1 =\int_a^b\vect f(\vect\gamma(t)) \cdot\vect n(\vect\gamma(t))\|\vect\gamma'(t)\|\,dt \end{equation*}
By our Definition 7.15 of line integrals we see that
\begin{equation} \int_{C}f_1\,dx_2-f_2\,dx_1 =\int_C\vect f\cdot\vect n\,ds\text{.}\tag{11.1} \end{equation}
Let \(\Delta s\) denote a very small line element. If \(\vect f\) models the motion of a fluid (direction and velocity of fluid at a given point) then \(\vect f\cdot\vect n\Delta s\) is the flux across \(\partial D\) through the line element \(\Delta s\text{,}\) and \(\vect f\cdot\vect n\Delta s\) is the approximate amount of fluid flowing across the boundary \(\partial D\) through \(ds\text{.}\) The situation is depicted in Figure 11.2. Note that the area of the parallelogram and the rectangle are the same.
Figure 11.2. Flux across a boundary element of \(\partial D\text{.}\) The parallelogram and the rectangle have the same area.
Hence the integral on the right hand side of (11.1) represents the total flux of \(\vect f\) across \(C\subset \partial D\text{.}\) Note that the above arguments are completely analogous to those used in Section 9.6 to find the flux of a vector field in space across a surface. Going back to the original formula we see that
\begin{equation*} \iint_D\Bigl(\frac{\partial}{\partial x_1}f_1 +\frac{\partial}{\partial x_2}f_2\Bigr)\,dx_1\,dx_2 =\int_{\partial D}\vect f\cdot\vect n\,ds \end{equation*}
is the total loss (or increase) of the fluid from \(D\) through \(\partial D\text{.}\) We make the following definition.

Definition 11.3. Divergence of a vector field.

If \(\vect f=(f_1,\dots,f_N)\) is a vector field in \(\mathbb R^N\) then
\begin{equation*} \divergence\vect f:=\frac{\partial}{\partial x_1}f_1+\frac{\partial}{\partial x_2} f_2+\dots +\frac{\partial}{\partial x_N} f_N \end{equation*}
is called the divergence of the vector field.

Remark 11.4.

Often the divergence of a vector field is written using the nabla operator in Definition 8.15. Formally, \(\divergence\vect f\) is the scalar product of \(\nabla\) with \(\vect f\text{,}\) and so we write
\begin{equation*} \nabla\cdot\vect f:=\divergence\vect f\text{.} \end{equation*}
We can now rewrite Green’s theorem. In that form it has a different name.
There is a direct generalisation of the Divergence Theorem to three or more dimensions. We will discuss it in Chapter 12.
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