Math S21a: Multivariable calculus Oliver Knill, Summer 2011
Lecture 22: Curl and Divergence
We have seen the curl in two dimensions: curl(F) = Qx−Py. By Greens theorem, it had b een
the average work of the field done along a small circle of radius raround the point in the limit
when the radius of the circle goes to zero. Greens theorem so has explained what the curl is. In
three dimensions, the curl is a vector:
The curl of a vector field ~
F=hP, Q, Riis defined as the vector field
curl(P, Q, R) = hRy−Qz, Pz−Rx, Qx−Pyi.
Invoking nabla calculus, we can write curl( ~
F) = ∇ × ~
F. Note that the third comp onent of the
curl is for fixed zjust the two dimensional vector field ~
F=hP, Qiis Qx−Py. While the curl
in 2 dimensions is a scalar field, it is a vector in 3 dimensions. In ndimensions, it would have
dimension n(n−1)/2. This is the number of two dimensional coordinate planes in ndimensions.
The curl measures the ”vorticity” of the field.
If a field has zero curl everywhere, the field is called irrotational.
The curl is often visualized using a ”paddle wheel”. If you place such a wheel into the field into
the direction v, its rotation speed of the wheel measures the quantity ~
F·~v. Consequently, the
direction in which the wheel turns fastest, is the direction of curl(~
F). Its angular velocity is the
length of the curl. The wheel could actually b e used to measure the curl of the vector field at any
point. In situations with large vorticity like in a tornado, one can ”see” the direction of the curl
near the vortex center.
In two dimensions, we had two derivatives, the gradient and curl. In three dimensions, there are
three fundamental derivatives, the gradient, the curl and the divergence.
The divergence of ~
F=hP, Q, Riis the scalar field div(hP, Q, Ri) = ∇ · ~
F=
Px+Qy+Rz.
The divergence can also be defined in two dimensions, but it is not fundamental.
The divergence of ~
F=hP, Qiis div(P, Q) = ∇ · ~
F=Px+Qy.
In two dimensions, the divergence is just the curl of a −90 degrees rotated field ~
G=hQ, −Pi
because div( ~
G) = Qx−Py= curl( ~
F). The divergence measures the ”expansion” of a field. If a
field has zero divergence everywhere, the field is called incompressible.
With the ”vector” ∇=h∂x, ∂y, ∂zi, we can write curl(~
F) = ∇× ~
Fand div( ~
F) = ∇· ~
F. Formulating
formulas using the ”Nabla vector” and using rules from geometry is called Nabla calculus. This
works both in 2 and 3 dimensions even so the ∇vector is not an actual vector but an operator.
The following combination of divergence and gradient often appears in physics:
∆f= div(grad(f)) = fxx +fyy +fzz .
It is called the Laplacian of f. We can write ∆f=∇2fbecause ∇ · (∇f) =
div(grad(f).
We can extend the Laplacian also to vector fields with
∆~
F= (∆P, ∆Q, ∆R) and write ∇2~
F.
Here are some identities:
div(curl( ~
F)) = 0.
curlgrad( ~
F) = ~
0
curl(curl( ~
F)) = grad(div( ~
F)−∆( ~
F)).
Proof. ∇ · ∇ × ~
F= 0.
∇ × ∇ ~
F=~
0.
∇ × ∇ × ~
F=∇(∇ · ~
F)−(∇ · ∇)~
F.
1Question: Is there a vector field ~
Gsuch that ~
F=hx+y, z, y 2i= curl( ~
G)?
Answer: No, because div( ~
F) = 1 is incompatible with div(curl ( ~
G)) = 0.
2Show that in simply connected region, every irrotational and incompressible field can be
written as a vector field ~
F= grad(f) with ∆f= 0. Proof. Since ~
Fis irrotational, there exists
a function fsatisfying F= grad(f). Now, div(F) = 0 implies divgrad(f) = ∆f= 0.
3Find an example of a field which is both incompressible and irrotational. Solution. Find
fwhich satisfies the Laplace equation ∆f= 0, like f(x, y) = x3−3xy2, then lo ok at its
gradient field ~
F=∇f. In that case, this gives
~
F(x, y) = h3x2−3y2,−6xyi.
