\(\newcommand{\curl}{\operatorname{curl}}\) \(\newcommand{\div}{\operatorname{div}}\) \(\newcommand{\grad}{\operatorname{grad}}\) \(\newcommand{\R}{\mathbb R}\)
\(\Leftarrow\) \(\Uparrow\) \(\Rightarrow\)
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.
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 \(n=3\):
Whenever we refer to the curl, we are always assuming that the vector field is \(3\) dimensional, since we are using the cross product.
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 \(C^2\).
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.
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:
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 \(\curl\).
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.
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.
Compute the curl and divergence of the following vector fields. The linear examples are recommended – they are both easy and instructive.
\(\mathbf F(x,y,z) = (x^2\cos y, zyx, e^{xy})\).
\(\mathbf F(x,y,z) = ( \sin yz, xz\cos yz, xy\cos yz)\).
Let \(A\) be the (antisymmetric) matrix \[ A =\left(\begin{array}{ccc} 0&-a&-b\\ a&0&-c\\ b&c&0 \end{array} \right) \] and let \(\mathbf F(\mathbf x) = A\mathbf x\).
Let \(S\) be the (symmetric) matrix \[ S =\left(\begin{array}{ccc} a&b&c\\ b&d&e\\ c&e&f\end{array} \right) \] and let \(\mathbf F(\mathbf x) = S\mathbf x\).
Let \(M\) be an arbitrary matrix \[ M =\left(\begin{array}{ccc} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{array} \right) \] and let \(\mathbf F(\mathbf x) = M\mathbf x\).
\(\mathbf F = (0,0,f(x,y,z))\), where \(f\) is a \(C^1\) function.
\(\mathbf F = (f(x,y), g(x,y), 0)\), where \(f\) and \(g\) are \(C^1\) functions.
Compute \(\Delta f\) for \(f:\R^3\setminus\left\{\bf 0\right\}\to \R\) defined by \(f(\mathbf x) = \frac 1{|\mathbf x|}\).
Prove one or more of the product rules for vector derivatives.
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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