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## 16.5 Divergence and Curl

Divergence and curl are two measurements of vector fields that are

very useful in a variety of applications. Both are most easily

understood by thinking of the vector field as representing a flow of a

liquid or gas;

that is, each vector in the vector field should be interpreted as a

velocity vector.

Roughly speaking, divergence

measures the tendency of

the fluid to collect or disperse at a point, and curl measures the

tendency of the fluid to swirl around the point. Divergence is a

scalar, that is, a single number, while curl is itself a vector. The

magnitude of the curl measures how much the fluid is swirling, the

direction indicates the axis around which it tends to swirl. These

ideas are somewhat subtle in practice, and are beyond the scope of

this course. You can find additional information on the web, for

example at

and

and in

many books including Div, Grad, Curl, and All That: An Informal Text on Vector Calculus,

by H. M. Schey.

Recall that if $f$ is a function, the gradient of $f$

is given by

$$\nabla f=\left\langle {\partial f\over\partial x}, {\partial

f\over\partial y}, {\partial f\over\partial z}\right\rangle. $$

A useful mnemonic for this (and for the divergence and curl, as it

turns out) is to let

$$\nabla = \left\langle{\partial \over\partial x}, {\partial

\over\partial y}, {\partial \over\partial z}\right\rangle, $$

that is, we pretend that $\nabla$ is a vector with rather odd looking

entries. Recalling that $\langle u, v, w\rangle a=\langle ua, va, wa\rangle$,

we can then think of the gradient as

$$\nabla f=\left\langle{\partial \over\partial x}, {\partial

\over\partial y}, {\partial \over\partial z}\right\rangle f =

\left\langle {\partial f\over\partial x}, {\partial

f\over\partial y}, {\partial f\over\partial z}\right\rangle, $$

that is, we simply multiply the $f$ into the vector.

The divergence and curl can now be defined in terms of this same odd

vector $\nabla$ by using the cross product and dot product.

The divergence of a vector field ${\bf F}=\langle f, g, h\rangle$ is

$$\nabla \cdot {\bf F} =

\left\langle{\partial \over\partial x}, {\partial

\over\partial y}, {\partial \over\partial z}\right\rangle\cdot

\langle f, g, h\rangle

= {\partial f\over\partial x}+{\partial

g\over\partial y}+{\partial h\over\partial z}. $$

The curl of $\bf F$ is

$$\nabla\times{\bf F} = \left|\matrix{{\bf i}&{\bf j}&{\bf k}\cr

{\partial \over\partial x}&{\partial

\over\partial y}&{\partial \over\partial z}\cr

f&g&h\cr}\right| =

\left\langle {\partial h\over\partial y}-{\partial g\over\partial z},

{\partial f\over\partial z}-{\partial h\over\partial x},

{\partial g\over\partial x}-{\partial f\over\partial y}\right\rangle. $$

Here are two simple but useful facts about divergence and curl.

Theorem 16. 5. 1

$\nabla\cdot(\nabla\times{\bf F})=0$.

$\qed$

In words, this says that the divergence of the curl is zero.

Theorem 16. 2

$\nabla\times(\nabla f) = {\bf 0}$.

That is, the curl of a gradient is the zero vector. Recalling that

gradients are conservative vector fields, this says that the curl of a

conservative vector field is the zero vector. Under suitable

conditions, it is also true that if the curl of $\bf F$ is $\bf 0$

then $\bf F$ is conservative. (Note that this is exactly the same test

that we discussed

in section 16. 3. )

Example 16. 3 Let ${\bf F} = \langle e^z, 1, xe^z\rangle$. Then

$\nabla\times{\bf F} = \langle 0, e^z-e^z, 0\rangle = {\bf 0}$.

Thus, $\bf F$ is conservative, and we can exhibit this directly by

finding the corresponding $f$.

Since $f_x=e^z$, $f=xe^z+g(y, z)$. Since $f_y=1$, it must be that

$g_y=1$, so $g(y, z)=y+h(z)$. Thus $f=xe^z+y+h(z)$ and

$$xe^z = f_z = xe^z + 0 + h'(z), $$

so $h'(z)=0$, i. e., $h(z)=C$, and $f=xe^z+y+C$.

$\square$

We can rewrite Green’s Theorem using these new ideas; these rewritten

versions in turn are closer to some later theorems we will see.

Suppose we write a two dimensional vector field in the

form ${\bf F}=\langle P, Q, 0\rangle$, where $P$ and $Q$ are functions

of $x$ and $y$. Then

$$\nabla\times {\bf F} =

\left|\matrix{{\bf i}&{\bf j}&{\bf k}\cr

P&Q&0\cr}\right|=

\langle 0, 0, Q_x-P_y\rangle, $$

and so $(\nabla\times {\bf F})\cdot{\bf k}=\langle 0, 0, Q_x-P_y\rangle\cdot

\langle 0, 0, 1\rangle = Q_x-P_y$. So Green’s Theorem says

$$\eqalignno{

\int_{\partial D} {\bf F}\cdot d{\bf r}&=

\int_{\partial D} P\, dx +Q\, dy = \dint{D} Q_x-P_y \, dA

=\dint{D}(\nabla\times {\bf F})\cdot{\bf k}\, dA. &

(16. 1)\cr}$$

Roughly speaking, the right-most integral adds up the curl (tendency

to swirl) at each point in the region; the left-most integral adds up

the tangential components of the vector field around the entire

boundary. Green’s Theorem says these are equal, or roughly, that the

sum of the “microscopic” swirls over the region is the same as the

“macroscopic” swirl around the boundary.

Next, suppose that the boundary $\partial D$ has a vector form

${\bf r}(t)$, so that ${\bf r}'(t)$ is tangent to the boundary, and

${\bf T}={\bf r}'(t)/|{\bf r}'(t)|$ is the usual unit tangent vector.

Writing ${\bf r}=\langle x(t), y(t)\rangle$ we get

$${\bf T}={\langle x’, y’\rangle\over|{\bf r}'(t)|}$$

and then

$${\bf N}={\langle y’, -x’\rangle\over|{\bf r}'(t)|}$$

is a unit vector perpendicular to $\bf T$, that is, a unit normal to

the boundary.

Now

$$\eqalign{

\int_{\partial D} {\bf F}\cdot{\bf N}\, ds&=

\int_{\partial D} \langle P, Q\rangle\cdot{\langle

y’, -x’\rangle\over|{\bf r}'(t)|} |{\bf r}'(t)|dt=

\int_{\partial D} Py’\, dt – Qx’\, dt\cr

&=\int_{\partial D} P\, dy – Q\, dx

=\int_{\partial D} – Q\, dx+P\, dy. \cr}$$

So far, we’ve just rewritten the original integral using alternate

notation. The last integral looks just like the right side of Green’s

Theorem (16. 4. 1) except that $P$ and $Q$ have

traded places and $Q$ has acquired a negative sign. Then applying

Green’s Theorem we get

$$

\int_{\partial D} – Q\, dx+P\, dy=\dint{D} P_x+Q_y\, dA=

\dint{D} \nabla\cdot{\bf F}\, dA. $$

Summarizing the long string of equalities,

\int_{\partial D} {\bf F}\cdot{\bf N}\, ds&=\dint{D} \nabla\cdot{\bf F}\, dA. 2)\cr}$$

Roughly speaking, the first integral adds up the flow across the

boundary of the region, from inside to out, and the second sums the

divergence (tendency to spread) at each point in the interior. The

theorem roughly says that the sum of the “microscopic” spreads is

the same as the total spread across the boundary and out of the region.

Exercises 16. 5

Sage knows how to compute divergence and curl.

Ex 16. 1

Let ${\bf F}=\langle xy, -xy\rangle$ and

let $D$ be given by $0\le x\le 1$, $0\le y\le 1$.

Compute $\ds\int_{\partial D} {\bf F}\cdot d{\bf r}$ and

$\ds\int_{\partial D} {\bf F}\cdot{\bf N}\, ds$.

(answer)

Ex 16. 2

Let ${\bf F}=\langle ax^2, by^2\rangle$ and

Ex 16. 3

Let ${\bf F}=\langle ay^2, bx^2\rangle$ and

let $D$ be given by $0\le x\le 1$, $0\le y\le x$.

Ex 16. 4

Let ${\bf F}=\langle \sin x\cos y, \cos x\sin y\rangle$ and

let $D$ be given by $0\le x\le \pi/2$, $0\le y\le x$.

Ex 16. 5

Let ${\bf F}=\langle y, -x\rangle$ and

let $D$ be given by $x^2+y^2\le 1$.

Ex 16. 6

Let ${\bf F}=\langle x, y\rangle$ and

Ex 16. 7

Prove theorem 16. 1.

Ex 16. 8

Prove theorem 16. 2.

Ex 16. 9

If $\nabla \cdot {\bf F}=0$, $\bf F$ is said to be incompressible. Show that any vector field

of the form ${\bf F}(x, y, z) = \langle f(y, z), g(x, z), h(x, y)\rangle$ is

incompressible. Give a non-trivial example.

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## Curl | mathematics | Britannica

Home

Science

Physics

Matter & Energy

curl, In mathematics, a differential operator that can be applied to a vector-valued function (or vector field) in order to measure its degree of local spinning. It consists of a combination of the function’s first partial derivatives. One of the more common forms for expressing it is: in which v is the vector field (v1, v2, v3), and v1, v2, v3 are functions of the variables x, y, and z, and i, j, and k are unit vectors in the positive x, y, and z directions, respectively. In fluid mechanics, the curl of the fluid velocity field (i. e., vector velocity field of the fluid itself) is called the vorticity or the rotation because it measures the field’s degree of rotation around a given point. This article was most recently revised and updated by William L. Hosch, Associate Editor.

Learn More in these related Britannica articles:

mathematics: Linear algebra

…the names div, grad, and curl, have become the standard tools in the study of electromagnetism and potential theory. To the modern mathematician, div, grad, and curl form part of a theory to which Stokes’s law (a special case of which is Green’s theorem) is central. The Gauss-Green-Stokes theorem, named…

principles of physical science: Nonconservative fields

…new function is needed, the curl, whose name suggests the connection with circulating field lines. …

fluid mechanics: Navier-stokes equation

…× v)—is sometimes designated as curl v [and ∇ × (∇ × v) is also curl curl v]. Another name for (∇ × v), which expresses particularly vividly the characteristics of the local flow pattern that it represents, is vorticity. In a sample of fluid that is rotating like a…

## Curl — from Wolfram MathWorld

The curl of a vector field, denoted or (the notation used in this work), is defined as the vector field having magnitude equal to the maximum “circulation”

at each point and to be oriented perpendicularly to this plane of circulation for

each point. More precisely, the magnitude of is the limiting

value of circulation per unit area. Written explicitly,

(1)

where the right side is a line integral around an infinitesimal region of area that is allowed to shrink to zero via

a limiting process and is the unit normal vector to this region.

If, then the field is said to be an irrotational

field. The symbol is variously known as “nabla”

or “del. ”

The physical significance of the curl of a vector field is the amount of “rotation” or angular momentum of the contents of given

region of space. It arises in fluid mechanics and elasticity theory. It is also fundamental

in the theory of electromagnetism, where it arises in two of the four Maxwell equations,

(2)

(3)

where MKS units have been used here, denotes the electric

field, is the magnetic field, is a constant

of proportionality known as the permeability of free space, is the current

density, and is another constant of proportionality

known as the permittivity of free space. Together with the two other of the Maxwell

equations, these formulas describe virtually all classical and relativistic properties

of electromagnetism.

In Cartesian coordinates, the curl is defined

by

(4)

This provides the motivation behind the adoption of the symbol for the curl,

since interpreting as the gradient

operator,

the “cross product” of the gradient operator

with is given by

(5)

which is precisely equation (4). A somewhat more elegant formulation

of the curl is given by the matrix operator equation

(6)

(Abbott 2002).

The curl can be similarly defined in arbitrary orthogonal curvilinear

coordinates using

(7)

and defining

(8)

as

(9)

(10)

The curl can be generalized from a vector field to

a tensor field as

(11)

where is the permutation

tensor and “;” denotes a covariant

derivative.

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## Frequently Asked Questions about what is the curl

### What is curl and divergence?

Roughly speaking, divergence measures the tendency of the fluid to collect or disperse at a point, and curl measures the tendency of the fluid to swirl around the point. Divergence is a scalar, that is, a single number, while curl is itself a vector.

### What is meant by curl of a vector?

The curl of a vector is always a vector quantity. The curl of a vector field provides a. measure of the amount of rotation of the vector field at a point. In general, the curl of any vector point function gives the measure of angular velocity at any. point of the vector field.

### What does curl mean in math?

Curl, In mathematics, a differential operator that can be applied to a vector-valued function (or vector field) in order to measure its degree of local spinning. It consists of a combination of the function’s first partial derivatives.