Differential Operators on Vector Fields

Section 5.3 Differential Operators on Vector Fields

Subsection 5.3.1 Definitions

A derivative of a function represented an account of how the function changes. The derivative was challenging to extend to scalar fields: new derivatives included partial derivatives, gradients and directional derivatives. The same is true for scalar fields. In this section, I'm going to introduce two important differential operators on vector fields. Each of them will capture something about how the vector field changes, but none of them is a simple, entire extention of the single-variable derivative.

Differential operator on vector fields are defined via the \(\nabla\) operator defined in Calculus III. This was a vector of partial differential operators. Let me remind you what this operator looked like in two, three and \(n\) dimensions.

\begin{align*} \nabla \amp = \left( \frac{\del}{\del x}, \frac{\del}{\del y} \right)\\ \nabla \amp = \left( \frac{\del}{\del x}, \frac{\del}{\del y}, \frac{\del}{\del z} \right)\\ \nabla \amp = \left( \frac{\del}{\del x_1}, \frac{\del}{\del x_2}, \ldots, \frac{\del}{\del x_n} \right) \end{align*}

So far, \(\nabla\) was used to define gradients. If \(f\) is a scalar field, then the gradient \(\nabla f\) is a vector field describing the direction of greatest change. Now, however, I can define new operations on vector fields using \(\nabla\text{.}\) The first is an operation that uses the cross product, therefore is specific to \(\RR^3\text{.}\)

Definition 5.3.1.

Let \(F : \RR^3 \rightarrow \RR^3\) be a vector field. The curl of \(F\) is the cross product of \(\nabla\) and \(F\text{.}\) Note that this outputs a new vector field, not a scalar field.

\begin{equation*} \nabla \times F = \left( \frac{\del F_3}{\del y} - \frac{\del F_2}{\del z}, \frac{\del F_1}{\del z} - \frac{\del F_3}{\del x}, \frac{\del F_2}{\del x} - \frac{\del F_1}{\del y}, \right) \end{equation*}

Curl measures the tendency of the vector field to cause local rotation. Using the fluid flow interpretation, if I drop an object in the fluid, it will flow along the integral curves of the vector field. However, as it flows along, it may also start spinning about an axis. Curl measures the tendency of the vector field to cause such a spin. (This is very different from global rotation. The paths of rotation themselves may be circular without actually causing the object itself to spin. Likewise, the paths can be totally straight but still cause local rotation. Don't confusing curving with spinning.)

Definition 5.3.2.

A vector field with zero curl is called irrotational.

Example 5.3.3.

Consider \(F(x,y,z) = (y,0,0)\text{.}\) This is a field which moves objects in the \(x\) direction, but the speed of movement varies with the \(y\) coordinate. The curl is \(\nabla \times F = (0,0,-1)\text{.}\) This field causes a clockwise rotation about the \(z\) axis; as particles in the fluid move in the \(x\) direction, they start spinning clockwise around a vertical axis.

Definition 5.3.4.

Let \(F: \RR^n \rightarrow \RR^n\) be a vector field in any dimension. The divergence of \(F\) is the dot product \(\nabla \cdot F\text{.}\) Note that this outputs a new scalar field, not a vector field.

\begin{equation*} \nabla \cdot F = \frac{\del F_1}{\del x_1} + \frac{\del F_2}{\del x_2} + \ldots + \frac{\del F_n}{\del x_n} \end{equation*}

Divergence measure the tendency of a vector field to diffuse. Thinking in terms of gaseous fluids, a positive divergence at a point means that the density of the gas is decreasing. Some directions of flow may be inward and some outward, but there is a net diffusion of the gas. If the divergence is negative, the density is increasing and there is a net gathering of the gas.

Definition 5.3.5.

A vector where where the divergence is zero is called incompressible.

Many liquids are incompressible, at least locally and under reasonable energy circumstances. Water is usually treated as an incompressible fluid. The major difference in fluid dynamics between gases and liquid is compressibility.

Divergence allows for a reinterpretation of the Laplacian.

Definition 5.3.6.

Let \(f\) be is a scalar field \(\RR^n \rightarrow \RR\text{.}\) It's Laplacian is the divergence of its gradient: \(\nabla \cdot (\nabla f) = \nabla^2 f\text{.}\) Note that this outputs a scalar field.

The Laplacian was mentioned in the Calculus III, since the input and output are both scalar fields. However, the intermediate state (the gradient) is a vector field, and the second part of the operation (the divergence) is a vector operation.

The Laplacian is very important for many differential equations in physics. As the divergence of the gradient field, it measures where the scalar field leads to a gathering or diffusion. In a potential energy field, the Laplacian measures the sources (attractors) and repellors which generate the field.

Subsection 5.3.2 Multivariable Differential Equations

There were two important differential equations introduced in Calculus III: the heat equation and the wave equation. Extending these DES to several dimensions uses the Laplacian. Let me state the single-variabl eversions of the equation.

\begin{align*} \frac{\del f}{\del t} \amp = \alpha \frac{\del^2 f}{\del x^2}\\ \frac{\del^2 f}{\del t^2} \amp = \alpha \frac{\del^2 f}{\del x^2} \end{align*}

The single varialbe heat equation or wave equation describe heat profile or wave displacement along a one-dimensional object: a string, wire or rod, perhaps. However, heat profile can vary along a two or three dimension object, as can wave displacement. Therefore, it would be nice to have a multivariable version of these equations. The solution is quite simple: the second derivative simply becomes the Laplacian. The Laplacian measure that higher-dimensional analogue of concavity that cause heat diffusion/wave acceleration in the single variable case.

\begin{align*} \frac{\del f}{\del t} \amp = \alpha \nabla^2 f\\ \frac{\del^2 f}{\del t^2} \amp = \alpha \nabla^2 f \end{align*}

In the wave equation, if \(\nabla^2 f = 0\) then \(\frac{\del^2 f}{\del t^2} = 0\text{.}\) Therefore, \(f\) has, at most, a linear dependence in \(t\text{.}\) Since the dependence in \(x\) is often sinusoidal, a common solution here is a standing wave or a wave with constant velocity. This (among other, more confusing reasons) leads to the terminology of harmonic functions.

Definition 5.3.7.

A scalar field \(f\) is harmonic if \(\nabla^2 f = 0\)

Subsection 5.3.3 Interaction between Vector Operations

As with all differential operators, the first property to establish is linearity.

Proposition 5.3.8.

Let \(f\) and \(g\) be scalar fields on \(\RR^3\text{,}\) and let \(F\) and \(G\) be vector fields on \(\RR^3\text{.}\) Let \(a\) and \(b\) be constants.

\begin{align*} \nabla (af \pm bg) \amp = a \nabla f \pm b \nabla g\\ \nabla \times (aF \pm bG) \amp = a (\nabla \times F) \pm b (\nabla \times G)\\ \nabla \cdot (aF \pm bG) \amp = a (\nabla \cdot F) \pm b (\nabla \cdot G)\\ \nabla^2 (af \pm bg) \amp = a \nabla^2 f \pm b \nabla^2 g \end{align*}

Since \(\nabla\) is a differential operator, it doesn't distribute over multiplication. However, there are several generalizations of the Leibniz rule.

Proposition 5.3.9.

Let \(f\) be a scalar field on \(\RR^3\) and let \(F\) and \(G\) be vector fields on \(\RR^3\text{.}\)

\begin{align*} \nabla \cdot (F \times G) \amp = (\nabla \times F) \cdot G - F \cdot (\nabla \times G)\\ \nabla \cdot (fF) \amp = f (\nabla \cdot F) + (\nabla f) \cdot F\\ \nabla \times (fF) \amp = f (\nabla \times F) + (\nabla f) \times F \end{align*}

In the first Leibniz rule, there is subtractioninstead of the expected addition. In general versions of the Leibniz rule, this negative sign is quite common. Many versions of the rule have \((-1)^k\) for some \(k\text{,}\) showing that it is equally easy to have a sum or difference.

There are two important results about composition of vector operations.

Proposition 5.3.10.

Let \(f\) be a scalar field on \(\RR^3\) and \(F\) a vector field on \(\RR^3\text{.}\) Then \(\nabla \times (\nabla f) = 0\text{:}\) the curl of a gradient is zero. Also, \(\nabla \cdot (\nabla \times F) = 0\text{:}\) the divergence of a curl is zero.

This proposition is a glimpse of a very general result in analysis. When starting with the right definitions, applying the same (or asimilar) differential operator twice in a row should give the zero operator. Obviously, this isn't always true: partial derivatives certainly don't satisfy this, and even for \(\nabla\text{,}\) \(\nabla \cdot \nabla f \neq 0\) for a general scalar field. Near the end of these notes, I will give a general structure for understanding why differential operators should compose to zero. For now, know that these two results for \(\nabla\) are not a coincidence.

To finish this section, I'll give some examples of using these new differential operators. None of the calculations are new (I'm doing partial derivative, dot product and cross product), but it is still worth seeing how they all fit together in this new setting. It is also worth reviewing the interpretations of curl and gradient for these examples.

Example 5.3.11.

Consider the scalar field \(f(x,y) = \ln (x^2 + y^2)\text{,}\) which is defined everywhere except the origin.

\begin{align*} \nabla f \amp = \left( \frac{2x}{x^2 + y^2}, \frac{2y}{x^2+y^2} \right)\\ \nabla^2 f \amp = \frac{2}{x^2 + y^2} - \frac{4x^2}{(x^2 + y^2)^2} + \frac{2}{x^2+y^2} - \frac{4x^2}{(x^2+y^2)^2} = 0 \end{align*}

This is a harmonic scalar field.

Example 5.3.12.

Consider the vector field \(F(x,y,z) = (-z,x,-y)\text{.}\) \(\nabla \times F = (-1,1,1)\text{,}\) so the vector field causes the same local rotation at all points, about the axis \((-1,1,1)\) (as a local axis direction). \(\nabla \cdot F = (0,0,0)\text{,}\) so the flow is incompressible. This is a good field to potentially model a liquid flow that induces a particular local rotation everywhere.

Example 5.3.13.

Consider the vector field \(F(x,y,z) = (x^2,y^2,z^2)\text{.}\) \(\nabla \times F = (0,0,0)\text{,}\) so the field is irrotational. The divergence is \(\nabla \cdot F = (2x +2y+2z)\text{.}\) In the positive octant, the flow accelerates away from the origin, so the rate of difussion increases away from the origin. In other octants, we may have negative divergence, reflecting the fact that the local vector field directions are always positive.

Example 5.3.14.

Consider (again) the vector field \(F(x,y,z) = (-y,x,0)\) \(\nabla \times F = (0,0,2)\text{,}\) which shows the creation of local rotation about the \(z\) axis. Note that this is local rotation, which is seperate from the global rotation of the integral curves about the origin. Also, \(\nabla \cdot F = 0\text{,}\) so the flow is incompressible. Though the flow rotates in circles and causes ojects to spin as them move, it neither collects or diffuses anywhere.

Example 5.3.15.

Consider the vector field \(F(x,y,z) = (\sin x, \cos y, 0)\text{.}\) \(\nabla \times F = (0,0,0)\text{,}\) so the flow is irrotational. This is interesting, given the trigonometric functions; trigonometry might cause us to expect spinning. \(\nabla \cdot F = \cos x - \sin y\text{,}\) so there are various areas of diffusion and collection. The trigonometric term here causes diffusion/collection, not local rotation.

Example 5.3.16.

Consider the vector field \(F(x,y,z) = \left( \frac{1}{x+y}, \frac{1}{x+y}, 0 \right)\text{.}\) \(\nabla \times F = \left(0,0, \frac{-1}{(x-y)^2} - \frac{-1}{(x+y)^2} \right)\text{:}\) this curl is always negative and local rotation is always about the \(z\) axis. \(\nabla \cdot F = \frac{-1}{(x+y)^2} + \frac{1}{(x-y)^2}\text{:}\) therefore, \(F\) collects when \(x\) and \(y\) are dissimilar and diffuses when \(x\) and \(y\) are quite close.