What Is The Curl

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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 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
$\nabla\cdot(\nabla\times{\bf F})=0$.
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 $f=xe^z+y+C$.
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
$$\nabla\times {\bf F} =
\left|\matrix{{\bf i}&{\bf j}&{\bf k}\cr
\langle 0, 0, Q_x-P_y\rangle, $$
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
\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)|}$$
is a unit vector perpendicular to $\bf T$, that is, a unit normal to
the boundary.
\int_{\partial D} {\bf F}\cdot{\bf N}\, ds&=
\int_{\partial D} \langle P, Q\rangle\cdot{\langle
y’, -x’\rangle\over|{\bf r}'(t)|} |{\bf r}'(t)|dt=
\int_{\partial D} Py’\, dt – Qx’\, dt\cr
&=\int_{\partial D} P\, dy – Q\, dx
=\int_{\partial D} – Q\, dx+P\, dy. \cr}$$
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
$\ds\int_{\partial D} {\bf F}\cdot{\bf N}\, ds$.
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.
Curl | mathematics | Britannica

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Curl | mathematics | Britannica

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.
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…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…
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…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

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,
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,
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
This provides the motivation behind the adoption of the symbol for the curl,
since interpreting as the gradient
the “cross product” of the gradient operator
with is given by
which is precisely equation (4). A somewhat more elegant formulation
of the curl is given by the matrix operator equation
(Abbott 2002).
The curl can be similarly defined in arbitrary orthogonal curvilinear
coordinates using
and defining
The curl can be generalized from a vector field to
a tensor field as
where is the permutation
tensor and “;” denotes a covariant
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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.

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