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Section 6.3 Divergence theorem in \(\mathbb{R}^n\)

We show that the divergence theorem holds in \(\mathbb{R}^n\text{,}\) not just in \(\mathbb{R}^3\text{.}\) It follows again from the generalized Stokes' theorem, but we need to rewrite it a little bit to see this.

Subsection 6.3.1 A divergence theorem in \(\mathbb{R}^n\text{?}\)

In the previous section, we showed that the generalized Stokes' theorem, in the particular case where \(\omega\) is a two-form on \(\mathbb{R}^3\) and \(M\) is a solid region \(E \subset \mathbb{R}^3\text{,}\) reduces to the divergence theorem in \(\mathbb{R}^3\text{,}\) which reads

\begin{equation*} \iiint_E (\boldsymbol{\nabla} \cdot \mathbf{F})\ dV = \iint_{\partial E} \mathbf{F} \cdot d \mathbf{S}. \end{equation*}

Contrary to Green's and Stokes' theorem, the divergence theorem involves the divergence of the vector field, not the curl. While the notion of curl of a vector field is not so easy to generalize to \(\mathbb{R}^n\text{,}\) the divergence can be generalized easily.

More precisely, let \(\mathbf{F}(x_1, \ldots, x_n) = (f_1, \ldots, f_n)\) be a smooth vector field on \(U \subseteq \mathbb{R}^n\text{,}\) with \(f_1, \ldots, f_n: U \to \mathbb{R}\) smooth functions. We can define the divergence of \(\mathbf{F}\) naturally as

\begin{equation*} \boldsymbol{\nabla} \cdot \mathbf{F} = \sum_{i=1}^n \frac{\partial f_i}{\partial x_i} = \frac{\partial f_1}{\partial x_1} + \frac{\partial f_2}{\partial x_2} + \ldots + \frac{\partial f_n}{\partial x_n}. \end{equation*}

Now suppose that \(E \subset \mathbb{R}^n\) is a closed bounded region that consists of a closed \((n-1)\)-dimensional space \(\partial E\) and its interior. The integral of \(\boldsymbol{\nabla} \cdot \mathbf{F}\) over \(E\) is defined naturally in calculus as a multiple (“\(n\)-tuple”) integral, which can be rewritten as an iterated integral if \(E\) is recursively supported. The “surface” integral over \(\partial E\) can also be generalized; since \(\partial E\) is a \((n-1)\)-dimensional subspace in \(\mathbb{R}^n\text{,}\) it has a well defined normal vector. If \(E\) is canonically oriented (choose the canonical ordered basis on \(\mathbb{R}^n\)), we say that the induced orientation on \(\partial E\) corresponds to an outward pointing normal vector, as for \(\mathbb{R}^3\text{.}\) A natural question then arise: does the divergence theorem generalize to any dimension? That is, is it true that

\begin{equation*} \underbrace{\int \cdots \int_E}_{\text{$n$ times}} (\boldsymbol{\nabla} \cdot \mathbf{F})\ dV_n = \underbrace{\int \cdots \int_{\partial E}}_{\text{$(n-1)$ times}} (\mathbf{F} \cdot \mathbf{n}) d V_{n-1} \, \end{equation*}

where the integral on the left-hand-side is an \(n\)-tuple integral over the region \(E \subset \mathbb{R}^n\text{,}\) and the right-hand-side is an integral of the vector field \(\mathbf{F}\) over the parametrized surface \(\partial E\) with normal vector pointing outward?

The answer is yes, and it again follows from the generalized Stokes' theorem. But we need to rewrite the generalized Stokes' theorem a little bit to see this.

Subsection 6.3.2 Rewriting the generalized Stokes' theorem

Let us recall the generalized Stokes' theorem from Theorem 6.1.1:

\begin{equation*} \int_M d \omega = \int_{\partial M} \omega, \end{equation*}

where \(M\) is an oriented \(n\)-dimensional manifold, \(\partial M\) its boundary, and \(\omega\) a \((n-1)\)-form.

Let us focus on a case similar to the previous section, where we take \(\partial E\) to be a closed \((n-1)\)-dimensional space in \(\mathbb{R}^n\) (such as a closed surface in \(\mathbb{R}^3\) in the previous section), and \(E\) to be the \(n\)-dimensional region of \(\mathbb{R}^n\) consisting of \(\partial E\) and its interior. We assign to \(E\) the canonical orientation, and to \(\partial E\) the induced orientation corresponding to a normal vector pointing outwards.

There is then a natural way of constructing a \((n-1)\)-form, using the Hodge star operator from Section 4.8. Let \(\eta\) be a one-form on \(U \subseteq \mathbb{R}^n\text{.}\) Then \(\star \omega\) is a \((n-1)\)-form on \(U \subseteq \mathbb{R}^n\text{,}\) by definition of the Hodge star. So we can rewrite the generalized Stokes' theorem for the one-form \(\eta\) as follows:

\begin{equation*} \int_E d(\star \eta) = \int_{\partial E} \star \eta. \end{equation*}

This is really the same generalized Stokes' theorem, but instead of writing it in terms of a \((n-1)\)-form \(\omega\text{,}\) we write it in terms of a one-form \(\eta\text{.}\)

Why would that be of any use? The advantage is that we can easily translate to vector field concepts for all \(\mathbb{R}^n\text{,}\) since we can establish a direct translation between one-forms and vector fields regardless of the dimension.

Subsection 6.3.3 The divergence theorem in \(\mathbb{R}^n\)

There is a natural dictionary between one-forms and vector fields in \(\mathbb{R}^n\text{.}\) Let \(\eta\) be a one-form on \(U \subseteq \mathbb{R}^n\text{.}\) We can write:

\begin{equation*} \eta = \sum_{i=1}^n f_i dx_i, \end{equation*}

where the \(f_i: U \to \mathbb{R}\text{,}\) for \(i=1,\ldots,n\text{,}\) are smooth functions. We can associate to this one-form the smooth vector field

\begin{equation*} \mathbf{F} = (f_1, f_2, \ldots, f_n) \end{equation*}

on \(U \subseteq \mathbb{R}^n\text{.}\)

We would like to rewrite our variant of the generalized Stokes' theorem as an integral theorem for the vector field \(\mathbf{F}\text{.}\) Let us first prove a lemma that will enable us to rewrite the left-hand-side of the generalized Stokes' theorem.

By definition of the Hodge star, we have:

\begin{equation*} \star \eta = \sum_{i=1}^n (-1)^{i+1} f_i dx_1 \wedge \cdots \widehat{dx _i} \wedge \cdots \wedge dx_n, \end{equation*}

where the hat notation means that we take the wedge product of all \(dx_j\)'s except the \(dx_i\text{.}\) Calculating the exterior derivative, we get:

\begin{align*} d(\star \eta) =\amp \sum_{i=1}^n (-1)^{i-1} \frac{\partial f_i}{\partial x_i} dx_i \wedge dx_1 \wedge \cdots \widehat{dx _i} \wedge \cdots \wedge dx_n\\ =\amp \sum_{i=1}^n \frac{\partial f_i}{\partial x_i} dx_1 \wedge \cdots \wedge dx_n\\ =\amp (\boldsymbol{\nabla} \cdot \mathbf{F}) dx_1 \wedge \cdots \wedge dx_n. \end{align*}

As for the right-hand-side of the generalized Stokes' theorem, we need to rewrite the integral \(\int_{\partial E} \star \eta\) in terms of vector calculus objects. \(\partial E\) is a closed \((n-1)\)-dimensional space in \(\mathbb{R}^n\text{.}\) We can think of it as a parametric space \(\alpha: D \to \mathbb{R}^n\) for some closed bounded region \(D \subset \mathbb{R}^{n-1}\text{,}\) like we did for parametric curves in \(\mathbb{R}^2\) and parametric surfaces in \(\mathbb{R}^3\text{.}\) In this we case, we can define the integral by pulling back using the parametrization. We claim that the following lemma holds:

We will not prove this statement in general; we will only prove it for parametric curves and surfaces. In fact, for parametric surfaces, this is basically the statement that was already proven in Corollary 5.6.5; indeed, what we have in this case is a surface integral in \(\mathbb{R}^3\text{,}\) and because by Table 4.1.11 we know that the vector field associated to the two-form \(\star \eta\) is the same as the vector field associated to the one-form \(\eta\text{,}\) the result of Corollary 5.6.5 still holds here.

Let us then show that it holds for parametric curves in \(\mathbb{R}^2\text{.}\) In this case, \(\eta = f\ dx + g\ dy\text{,}\) with associated vector field \(\mathbf{F}=(f,g)\text{,}\) and \(\star \eta = f\ dy - g\ dx\text{.}\) Let \(\alpha: [a,b] \to \mathbb{R}^2\) be a parametric curve representing the boundary curve \(\partial E\text{.}\) Thus we have:

\begin{equation*} \int_{\partial E} \star \eta = \int_{\alpha} \star \eta = \int_{[a,b]} \alpha^*(\star \eta). \end{equation*}

If we write \(\alpha(t) = (x(t), y(t))\text{,}\) the pullback is

\begin{equation*} \alpha^* (\star \eta) = \left( f(\alpha(t)) y'(t) - g(\alpha(t)) x'(t)\right)\ dt. \end{equation*}

Now, the tangent vector to the parametric curve is

\begin{equation*} \mathbf{T}(t) = (x'(t), y'(t)). \end{equation*}

The outward pointing normal vector is then

\begin{equation*} \mathbf{n}(t) = (y'(t), - x'(t) ), \end{equation*}

as the two vectors must be orthogonal, and the overall sign of the normal vector is fixed by requiring the it points outwards. We thus see that we can write

\begin{equation*} \alpha^* (\star \eta) = (\mathbf{F} \cdot \mathbf{n})\ dt, \end{equation*}

and

\begin{equation*} \int_{\partial E} \star \eta= \int_{[a,b]} (\mathbf{F} \cdot \mathbf{n}) d t. \end{equation*}

Putting this together, we see that our variant of the generalized Stokes' theorem gives rise to a generalization of the divergence theorem of the previous section that now holds in any dimension.

Remark 6.3.4.

In some textbooks, the divergence theorem in \(\mathbb{R}^2\) is simply called “another form of Green's theorem”. The reason is that it actually follows directly from Green's theorem. Recall that, given a vector field \(\mathbf{F} = (f,g)\) in \(\mathbb{R}^2\text{,}\) Green's theorem states that

\begin{equation*} \iint_D (\boldsymbol{\nabla} \times \mathbf{F}) \cdot \mathbf{e}_3\ dA = \int_{\partial D} (\mathbf{F} \cdot \mathbf{T})\ dt. \end{equation*}

The left-hand-side can be rewritten explicitly as

\begin{equation*} \iint_D (\boldsymbol{\nabla} \times \mathbf{F}) \cdot \mathbf{e}_3\ dA = \iint_D \left( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right)\ dA, \end{equation*}

while the right-hand-side can be rewriten as

\begin{equation*} \int_{\partial D} (\mathbf{F} \cdot \mathbf{T})\ dt = \int_{\partial D} \left( f(\alpha(t)) x'(t) + g(\alpha(t)) y'(t) \right)\ dt, \end{equation*}

where \(\alpha(t) = (x(t), y(t))\) is a parametrization of the curve \(\partial D\text{.}\)

Now if we consider a new vector field \(\mathbf{G} = (-g,f)\text{,}\) Green's theorem applied to \(\mathbf{G}\) is the statement that

\begin{equation*} \iint_D (\boldsymbol{\nabla} \times \mathbf{G}) \cdot \mathbf{e}_3\ dA = \int_{\partial D} (\mathbf{G} \cdot \mathbf{T})\ dt, \end{equation*}

which becomes, once written out explicitly,

\begin{equation*} \iint_D \left( \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} \right)\ dA = \int_{\partial D} \left( f(\alpha(t)) y'(t) - g(\alpha(t)) x'(t) \right)\ dt. \end{equation*}

But if we rewrite this expression in terms of the original vector field \(\mathbf{F} = (f,g)\text{,}\) we get

\begin{equation*} \iint_D (\boldsymbol{\nabla} \cdot \mathbf{F})\ dA = \int_{\partial D} (\mathbf{F} \cdot \mathbf{n})\ dt, \end{equation*}

which is the divergence theorem in \(\mathbb{R}^2\) for \(\mathbf{F}\text{!}\)

So Green's theorem and the divergence theorem in \(\mathbb{R}^2\) are really equivalent. But we prefer to call the later the divergence theorem in \(\mathbb{R}^2\) as it is the special case of the general divergence theorem in \(\mathbb{R}^n\text{.}\)

Remark 6.3.5.

Comparing Green's theorem and the divergence theorem in \(\mathbb{R}^2\text{,}\) it is interesting to note that the curl is related the tangential component of the vector field, while the divergence is related to the normal component. This is not a coincidence; if you recall from Section 4.5, the curl and divergence of vector fields are given a physical interpretation in terms of a moving fluid. The curl concerns whether a small sphere immersed in the fluid will rotate due to the fluid motion -- the rotation will be induced by the tangential component of the velocity field of the fluid on the surface of the sphere. The divergence concerns whether there is more fluid exiting than entering a small sphere immersed in the fluid -- this is mostly influenced by the normal component of the velocity field on the surface of the sphere. In fact, we can make this physical interpretation of the curl and div precise by applying the Stokes' and divergence theorem (respectively) to the small sphere, and take a limit where the volume of the sphere goes to zero. See for instance Section 4.4.1 in CLP 4 for this detailed calculation.