# Green's theorem

The term Green's theorem is applied to a collection of results that are really just restatements of the fundamental theorem of calculus in higher dimensional problems. The various forms of Green's theorem includes the Divergence Theorem which is called by physicists Gauss's Law, or the Gauss-Ostrogradski law. Stokes' theorem is another related result.

# Green's theorem for first order differential operators

## Fundamental theorem of Calculus

The classical definition of the definite integral of a function ${\displaystyle F(x)={\frac {df}{dx}}}$ on the closed interval ${\displaystyle [a,b]}$ as [1]

${\displaystyle \int _{a}^{b}F(x)\;dx=\int _{a}^{b}{\frac {df(x)}{dx}}\;dx=f(b)-f(a)}$

## Integration by parts

The second important result from calculus is the form of the Fundamental Theorem known as Integration by parts or partial integration.

${\displaystyle \displaystyle \int _{a}^{b}u\;dv={\Bigl .}u(x)v(x){\Bigr |}_{a}^{b}-\int _{a}^{b}v\;du.}$

## Green's theorem in 1D, first order differential operator

Suppose we have a second function ${\displaystyle g(x)}$ which is at least once differentiable and we write, applying integration by parts

${\displaystyle \int _{a}^{b}g(x)\left({\frac {df(x)}{dx}}\right)\;dx=\left.g(x)f(x)\right|_{b}^{a}-\int _{a}^{b}\left({\frac {d}{dx}}\right)(g(x))f(x)\;dx=\left.g(x)f(x)\right|_{a}^{b}+\int _{a}^{b}\left(-{\frac {d}{dx}}(g(x))\right)f(x)\;dx.}$

If we rewrite this with the integrations on the left hand side, we have

${\displaystyle \int _{a}^{b}\left[g(x)\left({\frac {d}{dx}}\right)f(x)-\left(-{\frac {d}{dx}}\right)(g(x))f(x)\right]\;dx={\Bigl .}g(x)f(x){\Bigr |}_{a}^{b}.}$

We note that the part in square brackets [] is an exact differential

${\displaystyle \left[g(x)\left({\frac {d}{dx}}\right)f(x)-\left(-{\frac {d}{dx}}\right)(g(x))f(x)\right]={\frac {d}{dx}}\left(g(x)f(x)\right)}$

More generally, if we have two operators ${\displaystyle A}$ and ${\displaystyle A^{\star }}$ and two functions ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ such that

${\displaystyle \int _{D}g(x)Af(x)\;dx=\int _{D}\left(A^{\star }g(x)\right)f(x)\;dx}$

then ${\displaystyle A^{\star }}$ is called the adjoint of ${\displaystyle A.}$

In the 1D, first order example, ${\displaystyle -{\frac {d}{dx}}}$ is the adjoint of ${\displaystyle {\frac {d}{dx}}.}$

## Green's theorem in 2D, first order differential operator

This result is important as it is a critical step in the proof of Cauchy's theorem in Complex Analysis.

Figure 1: Integration paths for the function ${\displaystyle P}$.
Figure 2: Integration paths for the function ${\displaystyle Q}$.

Here we follow standard texts, such as Spiegel (1964)[2] or Levinson and Redheffer (1970). [3]

A form of Green's theorem in two dimensions is given by considering two functions ${\displaystyle P(x,y)}$ and ${\displaystyle Q(x,y)}$ such that each of these functions is at least once differentiable inside and on a simple closed curve ${\displaystyle C}$ in a region of the ${\displaystyle x,y}$ plane.

If we consider a simple, closed curve ${\displaystyle C}$ and the integral over the area of bounded by ${\displaystyle C}$

${\displaystyle \int _{Area}\int {\frac {\partial P}{\partial y}}\;dxdy=\int _{a}^{b}\;dx\int _{Y_{1}(x)}^{Y_{2}(x)}{\frac {\partial P}{\partial y}}\;dy=\int _{a}^{b}\left(\left.P(x,y)\right|_{y=Y_{2}(x)}-\left.P(x,y)\right|_{y=Y_{1}(x)}\;dx\right)=-\oint _{C}P\;dx.}$

The last two equalities follow because we want to turn these two integrals, which both go from ${\displaystyle x=a}$ to ${\displaystyle x=b}$ into an integral over ${\displaystyle Y_{1}(x)}$ added to an integration from ${\displaystyle x=b}$ to ${\displaystyle x=a}$. This reverses the integration in the integral over ${\displaystyle Y_{2}(x)}$, accounting for the extra minus sign. See Figure 1.

Similarly we have

${\displaystyle \int _{Area}\int {\frac {\partial Q}{\partial x}}\;dxdy=\int _{e}^{f}\;dx\int _{X_{1}(y)}^{X_{2}(y)}{\frac {\partial Q}{\partial x}}\;dx=\int _{e}^{f}\left(\left.Q(x,y)\right|_{x=X_{2}(y)}-\left.Q(x,y)\right|_{x=X_{1}(y)}\;dy\right)=\oint _{C}Q\;dy.}$

Here the second integral which is over the curve ${\displaystyle X_{1}(y)}$ from ${\displaystyle y=e}$ to ${\displaystyle y=f}$ must be reversed, eliminating the minus sign on the second integral. See Figure 2.

In each case above, the integration order has been reversed, so as to make the integral in the counter-clockwise direction over the path ${\displaystyle C}$.

If we combine the two results, we have an equivalence between integrations of the derivatives of functions over an area and the line integral the functions over the boundary of that encloses the area

${\displaystyle \int _{Area}\int \left({\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\right)\;dxdy=\oint _{C}P\;dx+\oint _{C}Q\;dy.}$

This final result is called Green's theorem in two dimensions and is used in proofs of theorems in Complex Analysis.

## Green's theorem for second order differential operators

### Second order operators in 1D[4] [5]

We consider the case of a second derivative by performing integration by parts yields

${\displaystyle \int _{a}^{b}g(x)f^{\prime \prime }(x)\;dx=\left.g(x)f^{\prime }(x)\right|_{a}^{b}-\int _{a}^{b}g^{\prime }(x)f^{\prime }(x)\;dx.}$

Here the ${\displaystyle \prime }$ symbol represents differentiation with respect to ${\displaystyle x}$.

A second application of integration by parts applied to the integral on the right hand side yields

${\displaystyle \int _{a}^{b}g(x)f^{\prime \prime }(x)\;dx=\left.g(x)f^{\prime }(x)\right|_{a}^{b}-\int _{a}^{b}g^{\prime \prime }(x)f(x)\;dx-\left.g^{\prime }(x)f(x)\right|_{a}^{b}.}$

Rewriting so that the integral terms are on the left and the integrated terms are on the right, we obtain the familiar form of Green's theorem

${\displaystyle \int _{a}^{b}\left[g(x)f^{\prime \prime }(x)-g^{\prime \prime }(x)f(x)\right]\;dx=\left.\left[g(x)f^{\prime }(x)-g^{\prime }(x)f(x)\right]\right|_{a}^{b}.}$

## Green's theorem in higher dimensions

We consider a problem governed by a 2nd order partial differential operator, in a volume ${\displaystyle D}$ in ${\displaystyle n>2}$ dimensions with a boundary ${\displaystyle \partial D}$ of ${\displaystyle n-1}$ dimensions.

Such discussions are pertinent for many mathematical physics problems. The governing equation may be the Laplace equation, the Poisson equation, the heat or diffusion equation, or some form of the wave equation.

### The Divergence Theorem

The generalization of the fundamental theorem of calculus in higher dimensions is called the Divergence Theorem". The is also called Gauss's Law, or the Gauss-Ostrogradski Law. If we have an integral over the volume ${\displaystyle D}$ in ${\displaystyle R^{n}}$ bounded by the surface (really a hyper surface in dimensions greater than 3) ${\displaystyle \partial D}$ in ${\displaystyle R^{n-1}}$ then the following relation holds

${\displaystyle \int _{D}\mathbf {\nabla } \cdot \mathbf {Q} \;dV=\int _{\partial D}\mathbf {\hat {n}} \cdot \mathbf {Q} \;dS.}$

Here, ${\displaystyle Q}$ is a vector field, meaning that the integrand of the volume integral is an exact divergence. The quantity ${\displaystyle \mathbf {\hat {n}} }$ is the outward-pointing normal vector to the surface ${\displaystyle \partial D.}$ Note, in some materials the normal vector is chosen to be outward pointing, leading to a minus sign on the surface integral term.

### Green's theorem in ${\displaystyle n>2}$ dimensions, for 2nd order operators

Many mathematical physics problems are governed by second order partial differential operators. The governing equation may be the Laplace equation, the Poisson equation, the heat or diffusion equation, the Schroedinger equation, or some form of the wave equation.

#### Laplace equation

Potential field problems are governed by some form of the Laplace operator meaning that considering solutions to the Laplace operator is of importance. We may consider two solutions ${\displaystyle w(\mathbf {x} )}$ and ${\displaystyle v(\mathbf {x} }$ such that each are solutions to the Laplace equation

${\displaystyle \nabla ^{2}w(\mathbf {x} )=0}$

and

${\displaystyle \nabla ^{2}u(\mathbf {x} )=0.}$

If we form the integral

${\displaystyle \int _{D}w(\mathbf {x} )\nabla ^{2}u(\mathbf {x} )\;dV}$

and note that

${\displaystyle \nabla \cdot \left[w(\mathbf {x} )\nabla u(\mathbf {x} )\right]=\nabla w(\mathbf {x} )\cdot \nabla u(\mathbf {x} )+w(\mathbf {x} )\nabla ^{2}u(\mathbf {x} )}$

and thus

${\displaystyle w(\mathbf {x} )\nabla ^{2}u(\mathbf {x} )=\nabla \cdot \left[w(\mathbf {x} )\nabla u(\mathbf {x} )\right]-\nabla w(\mathbf {x} )\cdot \nabla u(\mathbf {x} )}$

We may write, using this identity that

${\displaystyle \int _{D}w(\mathbf {x} )\nabla ^{2}u(\mathbf {x} )\;dV=\int _{D}\nabla \cdot \left[w(\mathbf {x} )\nabla u(\mathbf {x} )\right]-\nabla w(\mathbf {x} )\cdot \nabla u(\mathbf {x} )\;dV.}$

Similarly, we may write

${\displaystyle \nabla \cdot \left[u(\mathbf {x} )\nabla w(\mathbf {x} )\right]=\nabla w(\mathbf {x} )\cdot \nabla u(\mathbf {x} )+u(\mathbf {x} )\nabla ^{2}w(\mathbf {x} )}$

and thus,

${\displaystyle \nabla w(\mathbf {x} )\cdot \nabla u(\mathbf {x} )=\nabla \cdot \left[u(\mathbf {x} )\nabla w(\mathbf {x} )\right]-u(\mathbf {x} )\nabla ^{2}w(\mathbf {x} ).}$

and substituting, we have:

${\displaystyle \int _{D}w(\mathbf {x} )\nabla ^{2}u(\mathbf {x} )\;dV=\int _{D}\nabla \cdot \left[w(\mathbf {x} )\nabla u(\mathbf {x} )\right]-\nabla w(\mathbf {x} )\cdot \nabla u(\mathbf {x} )\;dV=\int _{D}\nabla \cdot \left[w(\mathbf {x} )\nabla u(\mathbf {x} )\right]-\nabla \cdot \left[u(\mathbf {x} )\nabla w(\mathbf {x} )\right]+u(\mathbf {x} )\nabla ^{2}w(\mathbf {x} )\;dV.}$

Rearranging terms, we obtain

${\displaystyle \int _{D}\left\{w(\mathbf {x} )\nabla ^{2}u(\mathbf {x} )-u(\mathbf {x} )\nabla ^{2}w(\mathbf {x} )\right\}\;dV=\int _{D}\nabla \cdot \left[w(\mathbf {x} )\nabla u(\mathbf {x} )-u(\mathbf {x} )\nabla w(\mathbf {x} )\right]\;dV.}$

### Applying the divergence theorem

Applying the Divergence Theorem to the right hand side yields the familiar form of Green's theorem

${\displaystyle \int _{D}\left\{w(\mathbf {x} )\nabla ^{2}u(\mathbf {x} )-u(\mathbf {x} )\nabla ^{2}w(\mathbf {x} )\right\}\;dV=\int _{\partial D}\mathbf {\hat {n}} \cdot \left[w(\mathbf {x} )\nabla u(\mathbf {x} )-u(\mathbf {x} )\nabla w(\mathbf {x} )\right]\;dS.}$

Another variation may be obtained by defining the normal derivative as

${\displaystyle {\frac {\partial }{\partial n}}\equiv \mathbf {\hat {n}} \cdot \nabla }$

which allows us to write

${\displaystyle \int _{D}\left\{w(\mathbf {x} )\nabla ^{2}u(\mathbf {x} )-u(\mathbf {x} )\nabla ^{2}w(\mathbf {x} )\right\}\;dV=\int _{\partial D}\left[w(\mathbf {x} ){\frac {\partial u(\mathbf {x} )}{\partial n}}-u(\mathbf {x} ){\frac {\partial w(\mathbf {x} )}{\partial n}}\right]\;dS.}$

A typical physical sciences problem involves a second order partial differential equation, with boundary conditions. We may use Green's theorem to solve such problems.

For partial differential operators that are more general, consisting of the sum ${\displaystyle {\mathcal {L}}\equiv \nabla ^{2}+h(\mathbf {x} ),}$ then we may write

${\displaystyle \int _{D}\left\{w(\mathbf {x} ){\mathcal {L}}u(\mathbf {x} )-u(\mathbf {x} ){\mathcal {L}}w(\mathbf {x} )\right\}\;dV=\int _{\partial D}\left[w(\mathbf {x} ){\frac {\partial u(\mathbf {x} )}{\partial n}}-u(\mathbf {x} ){\frac {\partial w(\mathbf {x} )}{\partial n}}\right]\;dS.}$

Examples of self-adjoint operators are the Laplace and Poisson operators. A number of operators may be Fourier transformed in time to yield self-adjoint representations. These include the Schroedinger operator, the heat or diffusion operator, the scalar wave equation, and the Navier (elastic) wave equation.

In the more general case, the operator that makes the integrand into an exact divergence is not the same as the forward operator. This second operator is called the adjoint operator. An example of a class of second order non-self adjoint operators would be of the form ${\displaystyle a(\mathbf {x} )\nabla \cdot (b(\mathbf {x} )\nabla )+h(\mathbf {x} )}$ to yield the most general form of Green's theorem as

${\displaystyle \int _{D}\left\{w^{\star }(\mathbf {x} ){\mathcal {L}}u(\mathbf {x} )-u(\mathbf {x} ){\mathcal {L}}^{\star }w^{\star }(\mathbf {x} )\right\}\;dV=\int _{\partial D}\mathbf {\hat {n}} \cdot \mathbf {Q} \;dS.}$

Here the integrand of the volume integral is an exact divergence

${\displaystyle \left\{w^{\star }(\mathbf {x} ){\mathcal {L}}u(\mathbf {x} )-u(\mathbf {x} ){\mathcal {L}}^{\star }w^{\star }(\mathbf {x} )\right\}=\mathbf {\nabla } \cdot \mathbf {Q} }$

where the value of ${\displaystyle \mathbf {Q} }$ is specific to the problem.

## The application of Green's theorem

Green's theorem is the source of many powerful results in mathematical physics. These derive from an approach called loosely the Green's function method. This method permits a solution of a boundary value problem to be generated given knowledge of the Green's function of the governing equation and boundary conditions for the problem.

## References

1. Greenspan, Harvey Philip, and David J. Benney. Calculus: an introduction to applied mathematics. H, P. GREENSPAN, 1997.
2. Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).
3. Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.
4. Bleistein, N. (1984). Mathematical methods for wave phenomena. Academic Press.
5. Bleistein, N., J. K. Cohen, & J. W. Stockwell Jr., (2001). Mathematics of multidimensional seismic imaging, migration, and inversion. Springer Verlag.