Divergence, Curl and more Theorems#

Divergence & Curl#

Consider a 3D vector field \(\overrightarrow{\boldsymbol{F}}(x, y, z)=M(x, y, z) \mathbf{i}+N(x, y, z) \mathbf{j}+P(x, y, z) \mathbf{k}\).

The divergence of \(\overrightarrow{\boldsymbol{F}}\) is a scalar field: \(\quad \operatorname{div} \overrightarrow{\boldsymbol{F}}=\overrightarrow{\boldsymbol{\nabla}} \cdot \overrightarrow{\boldsymbol{F}}=\frac{\partial M}{\partial x}+\frac{\partial N}{\partial y}+\frac{\partial P}{\partial z}\)

The curl of \(\overrightarrow{\boldsymbol{F}}\) is a vector field: \(\quad \operatorname{curl} \overrightarrow{\boldsymbol{F}}=\overrightarrow{\boldsymbol{\nabla}} \times \overrightarrow{\boldsymbol{F}}=\left|\begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ M & N & P\end{array}\right|\)

Note that \(\vec{\nabla}=\mathbf{i} \frac{\partial}{\partial x}+\mathbf{j} \frac{\partial}{\partial y}+\mathbf{k} \frac{\partial}{\partial z}\) is the vector differential operator. (The symbol \(\nabla\) is sometimes read ‘nabla’ or ‘del’.)

Basically speaking, in terms of fluid flow, at a given point

  • \(\vec{\nabla} \cdot \overrightarrow{\boldsymbol{F}}\) tells us the rate at which the fluid spreads away from, or converges towards, the point (whether the point is a ‘source’ or a ‘sink’) - the rate of change of the fluid density at the point.

  • \(\overrightarrow{\boldsymbol{\nabla}} \times \overrightarrow{\boldsymbol{F}}\) gives a measure the rate of rotation of the fluid at the point - whether or how quickly fluid is swirling around the point.

  • If \(\vec{\nabla} \cdot \overrightarrow{\boldsymbol{F}}=\overrightarrow{\mathbf{0}}\) everywhere, the fluid is said to be incompressible.

  • If \(\vec{\nabla} \times \overrightarrow{\boldsymbol{F}}=\overrightarrow{\mathbf{0}}\) everywhere, the field is said to be curl-free or irrotational.

The Divergence Theorem#

Let \(E\) be a 3D region bounded by a closed surface \(S\). If \(\overrightarrow{\boldsymbol{F}}(x, y, z)\) is a smooth vector field defined on \(E\) then

\[ \iint_{S} \overrightarrow{\boldsymbol{F}} \cdot d \overrightarrow{\boldsymbol{S}}=\iiint_{E} \overrightarrow{\boldsymbol{\nabla}} \cdot \overrightarrow{\boldsymbol{F}} d V \]

Stokes’ Theorem#

Let \(\overrightarrow{\boldsymbol{F}}(x, y, z)\) be a smooth vector field defined on a surface \(S\) which is an oriented, piecewisesmooth, open surface bounded by a simple, piecewise-smooth, closed, positively-oriented curve \(C\) then

\[ \oint_{C} \overrightarrow{\boldsymbol{F}} \cdot d \overrightarrow{\boldsymbol{r}}=\iint_{S} \overrightarrow{\boldsymbol{\nabla}} \times \overrightarrow{\boldsymbol{F}} \cdot d \overrightarrow{\boldsymbol{S}} \]

Note: A given space curve \(C\) may be the boundary of many different surfaces \(S\). The surface integral will give the same value whichever surface \(S\) is used. (So for purposes of calculation, we may choose the simplest surface.)

Green’s Theorem in the Plane#

Let \(C\) be a piecewise smooth, simple closed curve with positive orientation and let \(D\) be the plane region bounded by \(C\). If \(M\) and \(N\) have continuous first order partial derivatives on an open region that contains \(D\), then

\[ \oint_{C} M d x+N d y=\iint_{D}\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right) d x d y \]

Optional further reading: Green’s theorem also provides a method for calculating an area by instead calculating a line integral around its boundary - see Thomas p.1011 or other textbooks.

Formal definitions (Optional)#

In 2D, divergence and curl are formally defined as:

\[ \vec{\nabla} \cdot \overrightarrow{\boldsymbol{F}}=\lim _{\left|A_{(x, y)}\right| \rightarrow 0}\left(\frac{1}{\left|A_{(x, y)}\right|} \oint_{C} \overrightarrow{\boldsymbol{F}} \cdot \hat{\boldsymbol{n}} d \overrightarrow{\boldsymbol{S}}\right) \]
\[ \vec{\nabla} \times \overrightarrow{\boldsymbol{F}}=\lim _{\left|A_{(x, y)}\right| \rightarrow 0}\left(\frac{1}{\left|A_{(x, y)}\right|} \oint_{C} \overrightarrow{\boldsymbol{F}} \cdot d \overrightarrow{\boldsymbol{r}}\right) \]

where \(A_{(x, y)}\) represents some region around the point \((x, y)\).

From these definitions we can understood divergence (curl) as the limit of the average value of the flux (circulation) over a region around a point as we let the region get smaller and smaller. - See Khan Academy website or advanced textbooks.