5.4 Div, Grad, Curl

5.4 Div, Grad, Curl

  1. Definitions
  2. Identities of Vector Derivatives
  3. Problems

\(\Leftarrow\)  \(\Uparrow\)  \(\Rightarrow\)

Definitions

Recall that for an open subset \(U\subseteq \R^n\) and \(f:U\to \R\) a \(C^1\) function, the gradient of \(f\) is \(\nabla f = (\partial_1 f, \ldots, \partial_n f).\) We are already very familiar with this. We will also call this grad \(f\), and may write \(\grad(f)\). Along with the total derivative \(D\mathbf F\), the vector fields that we work with have other derivatives that are useful.

For \(U\) an open subset of \(\R^n\) and \(\mathbf F:U\to \R^n\) a vector field of class \(C^1\), the divergence of \(\mathbf F\) is \[ \div \mathbf F = \nabla \cdot \mathbf F = \partial_1 F_1+ \cdots +\partial_n F_n. \]

This will give a real-valued function, corresponding to the trace of \(D\mathbb F\).

The definitions of \(\grad\) and \(\div\) make sense in \(\R^n\) for any \(n\). Our next definition only makes sense when \(n=3\):

For \(U\) an open subset of \(\R^3\) and \(\mathbf F:U\to \R^3\) a vector field of class \(C^1\), then the curl of \(\mathbf F\) is \[ \curl \mathbf F = \nabla \times\mathbf F = \left[ \partial_2F_3 - \partial_3 F_2, \, \partial_3F_1 - \partial_1 F_3, \, \partial_1F_2 - \partial_2 F_1\right]. \]

Whenever we refer to the curl, we are always assuming that the vector field is \(3\) dimensional, since we are using the cross product.

Identities of Vector Derivatives

Composing Vector Derivatives

Since the gradient of a function gives a vector, we can think of \(\grad f: \R^3 \to \R^3\) as a vector field. Thus, we can apply the \(\div\) or \(\curl\) operators to it. Similarly, \(\div F\) gives a function, so we can apply \(\grad\) to it, and \(\curl F\) gives a vector field, so we can apply \(\div\) or \(\curl\) to it. This gives five possible compositions of derivatives.

Something interesting happens with \(\curl(\grad f)\) and \(\div(\curl \mathbf F)\) when \(f\) and \(\mathbf F\) are \(C^2\).

Spoiler
Try the computations first We have \(\curl(\grad f) = {\bf 0}\) whenever \(f\) is \(C^2\), and \(\div(\curl \mathbf F) = 0\) whenever \(\mathbf F\) is \(C^2\).
Both of these are straightforward to verify, and both of them use Clairault’s theorem, which explains why we require them to be \(C^2\).

If we arrange \(\div, \grad, \curl\) as indicated below, then following any two successive arrows yields \(0\) (or \(\bf 0\)). \[ \text{functions} \quad\overset{\grad} { \xrightarrow{\hspace{1cm}}} \quad \text{vector fields} \quad \overset{\curl} {\xrightarrow{\hspace{1cm}}} \quad \text{vector fields} \quad \overset{\div} {\xrightarrow{\hspace{1cm}}} \quad \text{functions} . \] The remaining three compositions are also interesting, and they are not always zero.

For a \(C^2\) function \(f:\R^n\to \R\), the Laplacian of \(f\) is \(\div(\grad f) = \sum_{j=1}^n \partial_{jj}f\)

The Laplacian of \(f\) is usually denoted \(\Delta f\) or \(\nabla^2 f\). The former notation is used more often by mathematicians, and the latter by physicists and engineers. The Laplacian appears throughout mathematics, mathematical physics, chemistry, and also in many areas of applied mathematics, including mathematical finance.

The other possible two combinations of vector derivatives are \(\grad\div\) and \(\curl\curl\). These are related to the vector Laplacian:

For a \(C^2\) vector field \(\mathbf F:U\to \R^3\), with \(U\) an open subset of \(\R^3\), the vector Laplacian of \(\mathbf F\) is \[ \Delta \mathbf F = (\Delta F_1, \Delta F_2, \Delta F_3). \] where \((F_1,F_2,F_3)\) are the components of \(\mathbf F\).

Another straightforward calculation will show that \(\grad\div \mathbf F - \curl\curl \mathbf F = \Delta \mathbf F\).

The vector Laplacian also arises in diverse areas of mathematics and the sciences. The frequent appearance of the Laplacian and vector Laplacian in applications is really a testament to the usefulness of \(\div, \grad\), and \(\curl\).

Product Rules

Since there are several ways of multiplying functions and vector fields and then several kinds of vector derivatives, there are a number of different product rules for vector derivatives.

Suppose \(f, g:\R^n\to R\) are \(C^1\) functions, and \(\mathbf F, \mathbf G:\R^n\to\R^n\) are \(C^1\) vector fields. Then the following product rules hold (where \(n=3\) if \(\curl\) is used): \[ \begin{aligned} \grad (fg) &= f \grad g + g \grad f \\ \div (f\mathbf G) &= f\div \mathbf G + (\grad f) \cdot \mathbf G \\ \curl (f\mathbf G) &= f \curl \mathbf G +(\grad f) \times \mathbf G \\ \div(\mathbf F\times \mathbf G)&= \mathbf G\cdot \curl \mathbf F - \mathbf F\cdot \curl \mathbf G\\ \curl(\mathbf F\times \mathbf G)&= (\mathbf G\cdot \nabla)\mathbf F + (\div \mathbf G)\mathbf F+ (\mathbf F\cdot \nabla)\mathbf G + (\div \mathbf F) \mathbf G \\ \grad(\mathbf F\cdot \mathbf G) &= (\mathbf G\cdot \nabla)\mathbf F + \mathbf G\times \curl \mathbf F - (\mathbf F\cdot \nabla)\mathbf G - \mathbf F\times \curl \mathbf G \end{aligned} \] where \[ (\mathbf F\cdot \nabla)\mathbf G = \sum_{j=1}^3 F_j \partial_j \mathbf G, \qquad (\mathbf G\cdot \nabla)\mathbf F = \sum_{j=1}^3 G_j \partial_j \mathbf F. \]

Committing these product rules to memory may not be a good use of brainpower, but you should know that they exist, and if you come upon a situation when one or more of these is needed, you should be able to recognize that and look up the relevant identity.

ProofIn every case these can be proved by laboriously writing out both sides and checking to see that they are equal. This is recommended at most once per lifetime.

Problems

Basic

  1. Compute the curl and divergence of the following vector fields. The linear examples are recommended – they are both easy and instructive.

  2. Compute \(\Delta f\) for \(f:\R^3\setminus\left\{\bf 0\right\}\to \R\) defined by \(f(\mathbf x) = \frac 1{|\mathbf x|}\).


Hint \(|\mathbf x| = (|\mathbf x|^2)^{1/2}\)

Advanced

  1. Prove one or more of the product rules for vector derivatives.

  2. Prove that if \(f\) and \(g\) are \(C^2\) functions of \(3\) variables, then \(\div(\grad f \times \grad g) = 0\).

\(\Leftarrow\)  \(\Uparrow\)  \(\Rightarrow\)

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