Theorem 6.5.1.
A source-free vector field in the plane is also divergence-free.
That is, “\(\oint_C \vector{P, Q} \cdot \N ds = 0\) implies \(P_x + Q_y = 0\)”.
Section 6.5 Divergence and Curl
References.
- OpenStax Calculus Volume 3, Section 6.5.
1
openstax.org/books/calculus-volume-3/pages/6-5-divergence-and-curl - Calculus, Early Transcendentals by Stewart, Section 16.5.
Topics.
Subsection 6.5.1 Divergence
Subsubsection 6.5.1.1 The Divergence of a Two Dimensional Vector Field
For two dimensional vector fields, the object \(\nabla\) or \(\del\) introduced in Equation (6.1.4) of Section 6.1 is formally the vector \(\ds\del = \veci \frac{\partial}{\partial x} + \vecj \frac{\partial}{\partial y}\text{,}\) suggesting the dot product
\begin{equation*}
\del \cdot \vec{F}
= \left( \veci \frac{\partial}{\partial x} + \vecj \frac{\partial}{\partial y} \right) \cdot (\veci P + \vecj Q)
= \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}
\end{equation*}
This is the two dimensional divergence, also denoted \(\Div~\vec{F}\text{,}\) so the flux form of Green’s Theorem Theorem 6.4.3 can be written as the Two Dimensional Divergence Theorem
\begin{equation}
\oint_{\partial D} \vec{F} \cdot \N\ ds
= \iint\limits_D \del \cdot \vec{F} \, dA\tag{6.5.1}
\end{equation}
For the example of \(\vec{F}\) describing fluid velocity, \(\del \cdot \vec{F}\) is related to “net outflow”, underlying the name “divergence”.
Subsubsection 6.5.1.2 Divergence-free fields
Proof.
From Theorem 6.4.6 a source-free field has a stream function \(g\text{:}\) \(P = g_y\) and \(Q=-g_x\text{.}\) Thus Clairault’s Theorem gives
\begin{equation*}
P_x + Q_y = (g_y)_x + (-g_x)_y = 0\text{.}
\end{equation*}
As we have seen many times before, the converse is also true when the domain is also simply connected:
Theorem 6.5.2. Divergence Test for Source-Free Vector Fields.
A continuous and differentiable vector field \(\vec{F} = \vector{P, Q}\) on a simply connected domain is source-free if and only if it is divergence-free.
Proof.
The main idea of the proof is conecting to the analogous Theorem 6.3.8 by the strategy of “rotating by a quarter turn” seen in the proof of The Flux form of Green’s Theorem (6.4.4).
One important example is that magnetic fields are always descibed by divergence-free vector fields.
Subsubsection 6.5.1.3 The Divergence of a Three Dimensional Vector Field
For a vector field in three dimensions, \(\vec{F} = P \veci + Q \vecj + R \veck\text{,}\) the divergence of F is
\begin{equation*}
\Div~\vec{F} = \del \cdot \vec{F}
= \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}+ \frac{\partial R}{\partial z}.
\end{equation*}
The extension of Equation (6.5.1) to three dimensions is the Divergence Theorem 6.8.1 to be seen in Section 6.8.
Subsection 6.5.2 Curl
The first, “circulation” form of Green’s Theorem Theorem 6.4.2 suggests that the quantity \(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\) integrated measures (anti-clockwise) rotation in the vector field, perpendicur to the plane, and so around the z-azis.
It is useful to think of this as a vector quantity
\begin{equation*}
\left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \veck
\end{equation*}
and for vector fields in space, extending this to corresponding measures of rotation with respect to the \(x\) and \(y\) axes. This lead to the quantity sometimes called the rotation, \(\Rot\ \vec{F}\) when the vector field describes the velocity in a fluid, but now more often call the curl of a vector field on \(\reals^3\text{:}\)
Definition 6.5.3.
For a differentiable vector field \(\vec{F} = P\veci + Q\vecj + R\veck\) on \(\reals^3\) its curl is given by
\begin{equation}
\Curl~\vec{F}
= \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \veci + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \vecj + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \veck\tag{6.5.2}
\end{equation}
This is an important quantity in the description not only of fluid flow but also electro-magnetic fields, and it is related to whether the vector field is conservative.
For example, with a 2D vector field \(\vec{F} = P(x,y)\veci + Q(x,y)\vecj\text{,}\) this is just \(\Curl~\vec{F} = \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \veck\text{,}\) and so the condition \(\Curl~\vec{F} = \vec{0}\) is the cross-partials condition of Section 6.3 related to \(\vec{F}\) being conservative.
Using this 2D case of the curl, the original form of Green’s Theorem in Equation Theorem 6.4.2 can be put in the form
\begin{equation}
\oint_{\partial D} \vec{F} \cdot d\vec{r} = \oint_{\partial D} \vec{F} \cdot \vec{T} ds
= \iint\limits_D (\Curl~\vec{F}) \cdot \veck \ dA\tag{6.5.3}
\end{equation}
This is sometimes also called the planar version of Stokes’ Theorem 6.7.2 to be seen in Section 6.7.
Subsubsection 6.5.2.1 Curl and Rotation in a Fluid
The name “curl” refers to a measure of rotation. For example, the vector field \(\vec{F} = -y\veci + x \vecj\) describes velocity of a fluid (say) going anti-clockwise around the \(z\)-axis: it has \(\Curl~\vec{F} = 2\veck\) in which the direction \(\veck\) indicates the axis of rotation, the positive value indicates anti-clockwise direction as viewed from “above” down that \(z\)-axis, and the uniform magnitude indicating the uniform angular rate of rotation (which has period \(2\pi\)).
Subsubsection 6.5.2.2 A Short-hand Notation For the Curl
The formal \(\del\) short-hands seen for the gradient \(\Grad f = \del f\) and the divergence \(\Div f = \del \cdot \vec{F}\) also have a counterpart for the curl, via the formal cross product:
\begin{align*}
\del \times \vec{F}\\
\amp= \left| \begin{array}{ccc}
\veci \amp \vecj \amp \veck
\\
\frac{\partial}{\partial x} \amp \frac{\partial}{\partial y} \amp \frac{\partial}{\partial z}
\\
P \amp Q \amp R
\end{array} \right|\\
\amp= \veci \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)
+ \vecj \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right)
+ \veck \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\\
\amp= \Curl~\vec{F}.
\end{align*}
Hence we sometimes use \(\del \times \vec{F}\) as mnemonic notation for \(\Curl~\vec{F}\text{.}\)
Subsection 6.5.3 Some Connections Between Div, Curl and Grad
Subsubsection 6.5.3.1 Gradients are “Curl Free”, and vice versa on Simply Conected Domains
The curl is involved in the 3D version of the mixed partials condition in Section 6.3:
Theorem 6.5.4.
- The curl of a conservative vector field \(\vec{F} = \Grad f\) vanishes: that is\begin{equation*} \Curl(\Grad f) = \del \times (\del f) = (\del \times \del) f = \vec{0}. \end{equation*}(The final form is given as a mnemonic: if you think of \(\del\) as being a true vector, its cross product with itself should be zero.)
- On a simply conected domain, the converse is also true: if \(\Curl\ \vec{F} = \vec{0}\) then \(\vec{F}\) is a gradient.
Proof of part (1).
The verification of the first half is a straightforward calculation using Clairaut’s Theorem: it is like three versions of the mixed partials condition for a conservative vector field in \(\reals^2\) seen in Section 6.3.
The proof of part (2) is postponed to Section 6.7.
The proof of the converse (2) is difficult; the main ideas will be sketched in Subsection 6.7.4, using Stokes’ Theorem from Section 6.7.
As a hint, in the plane this is the conclusion of Theorem 6.3.8.
Subsubsection 6.5.3.2 Curls are “Divergence Free”
Theorem 6.5.5.
If \(\vec{F} = P \veci + Q \vecj + R \veck\) is a vector field on \(\reals^3\) and all components have continuous second derivatives, then
\begin{equation*}
\Div~\Curl~\vec{F} = \del \cdot (\del \times \vec{F}) = 0.
\end{equation*}
Proof.
This can again be verified by straightforward calculation and using Clairaut’s Theorem.
Subsubsection 6.5.3.3 The Remaining Pairs: The Laplacian, etc.
There are three combinations of two of these operators that are not automatically zero, of which the most important is
\begin{equation*}
\Div\ \Grad f = \del \cdot \del f
= \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}
\end{equation*}
called the Laplacian of \(f\). This is often abbreviated as \(\del^2 f\) or \(\Delta f\text{.}\) For functions of two variables this is
\begin{equation}
\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}\tag{6.5.4}
\end{equation}
This operation \(\Delta\) is called the Laplace operator, with the symbolic form
\begin{equation}
\Delta = \del^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\text{.}\tag{6.5.5}
\end{equation}
The Laplacian of a vector field is defined as the vector field got by applying the Laplace operator to each component in turn; that is:
\begin{equation*}
\Delta \vector{P, Q, R} = \Delta P \ \veci + \Delta Q \ \vecj + \Delta R \, \veck
\end{equation*}
The two other non-trivial combinations combine to give
\begin{equation*}
\Curl\ \Curl\ \vec{F} = \Grad\ \Div\ \vec{F} - \Div\ \Grad \vec{F}\text{;}
\end{equation*}
that is,
\begin{equation}
\del \times (\del \times \vec{F}) = \del(\del\cdot \vec{F}) - \Delta \vec{F}\text{.}\tag{6.5.6}
\end{equation}
Subsubsection 6.5.3.4 Summary
Here are formulas for all the combinations of Div, Curl and Grad that make sense:
- \(\displaystyle \Div\ \Grad f = \del\cdot\del f = \del^2 f = \Delta f = f_{xx} + f_{yy} + f_{xx}\)
- \(\displaystyle \del \times (\del \times \vec{F}) = \del(\del\cdot \vec{F}) - \Delta \vec{F}\)
- \(\displaystyle \Curl\ \Grad f = \del\times \del f = \vec{0}\)
- \(\displaystyle \Div\ \Curl\ \vec{F} = \del\cdot (\del\times \vec{F}) = 0\)
Subsection 6.5.4 Some Fundamental Differential Equations of Physics
The Laplacian arises in several of the fundamental equations for physics:
\begin{align*}
\text{Laplace's Equation} \amp\amp \Delta u
\amp = 0\\
\text{The Poisson Equation} \amp\amp \Delta u \amp = f,\\
\text{The Heat Equation} \amp\amp \frac{\partial u}{\partial t} \amp = \Delta u,\\
\text{The Wave Equation} \amp\amp \frac{\partial^2 u}{\partial t^2} \amp = \Delta u,\\
\text{and}\\
\text{The Schrödinger Equation} \amp\amp i \hbar \frac{\partial \psi}{\partial t} \amp = -\frac{\hbar^2}{2m}\Delta \psi + V(\vec{x})\psi
\end{align*}
used in the description of fluid motion, electric fields, heat conduction, waves in water and in electro-magnetic fields, and in quantum mechanics.
Study Guide.
Study Section 6.5 of OSC3; in particular
8
openstax.org/books/calculus-volume-3/pages/6-5-divergence-and-curl- The definitions of divergence and of curl.
- The concept of a source-free vector field: a vector field \(\vec{F} = \vector{P(x,y),Q(x,y)}\) for which there is a stream function \(g(x,y)\) for which \(P = g_y, Q = -g_x\) from Subsubsection 6.4.4.1, and its connection to a field being divergence-free: \(\del \cdot \vec{F} = 0\text{;}\) see Theorems 14 and 15.
- Theorem 16: \(\Div\ \Curl\ \vec{F} = 0\text{,}\) or \(\del \cdot (\del \times \vec{F}) = 0\) (as if \(\del\) were a real vector.)
- Theorem 17, which can be restated as \((\del \times \del) f = 0\) (again as if \(\del\) were a real vector.)
- Theorem 18, a restatement of the cross-partial condition of Theorem 6.3.8 in Section 6.3.
- Examples 48–50, 52–56, and the Checkpoints following each.
- The T/F Exercises 207–211 and one or several exercises from each of the following ranges: 212–221, 222–231, 256, 257, 258.
