Video - Divergence Theorem: The Simplest Explanation Every Physicist Should Know

Here the two formulas, called Green's identities, are derived using the Divergence theorem. Green's identities are useful identities for converting integrals with gradients and divergences into integrals with normal derivatives. They are used, for example, in electrostatics to calculate electric potentials.

Consider a vector field \(\boldsymbol{F}(\boldsymbol{r})\) (for example, it could be an electric field) that depends on the spatial coordinate \(\boldsymbol{r} = (x,y,z)\) and is defined as follows:

Vector field as product of a scalar function and gradient

Here \(\Phi(\boldsymbol{r})\) and \( \Psi(\boldsymbol{r}) \) are smooth scalar functions (e.g. they could be two electric potential functions). And the symbol \(\nabla\) is the nabla operator applied to the scalar function \( \Psi(\boldsymbol{r}) \). The result is the gradient of the scalar function: \(\nabla \, \Psi(\boldsymbol{r})\). In the following we do not write the dependence on \(\boldsymbol{r}\) to make the equations a bit more compact. Of course, the corresponding functions still depend on \(\boldsymbol{r}\).

We want to apply the Divergence theorem to the vector field \(\boldsymbol{F}\). The Divergence theorem states the following:

Here \(V\) is any volume and \(A\) is the associated closed surface of the enclosed volume. Using the Divergence theorem, we can convert a volume integral (left side of 2) to a surface integral (right side of 2) and vice versa.

As you can see, the divergence of vector field \( \nabla \cdot \boldsymbol{F} \) occurs on the left side of the Divergence theorem. Therefore, the first step we want to take is to form the divergence of our vector field \(\boldsymbol{F}\) defined in Eq. 1:

Here \(\nabla^2\) is the Laplace operator, that is the sum of the second derivatives with respect to the spatial coordinates \(x\), \(y\) and \(z\). In the second step, we have exploited the following identity for divergence:

Divergence of the product of a scalar function with a vector field

Formula anchor$$ \begin{align} \nabla ~\cdot~ (u \, \boldsymbol{v}) ~=~ u \, \nabla \cdot \boldsymbol{v} ~+~ \boldsymbol{v} \cdot \nabla u \end{align} $$

Now we insert the divergence field 3 into the left side of the Divergence theorem 2:

Divergence field inserted into the Divergence theorem

Next, we express the area element \(\text{d}\boldsymbol{a}\) with the area normal vector \(\boldsymbol{n} \). This normal vector is perpendicular to the area element and points in the same direction as \(\text{d}\boldsymbol{a}\) element:

Area element with the normal vector expressed in the Divergence theorem

Here \(\nabla \, \Psi \cdot \boldsymbol{n}\) is the directional derivative of \(\Psi\). It is the component of the gradient field \(\nabla \, \Psi\) in the direction of \(\boldsymbol{n}\) on the surface \(A\) (also called normal derivative). The normal derivative is also notated as follows:

Then we subtract from the 1st Green's identity the swapped version 11. Thus \( \nabla \Phi ~\cdot~ \nabla \Psi \) is eliminated, since divergence operation \(~\cdot~\) is commutative. What remains is:

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