Section 8.4 Closed Vector Fields in Space
If \(N=3\) and and \(\vect f=(f_1,f_2,f_3)\) is a vector field then \(\vect f\) is closed if
\begin{align*}
\frac{\partial}{\partial x_2}f_3(\vect x)
\amp=\frac{\partial}{\partial x_3}f_2(\vect x)\\
\frac{\partial}{\partial x_3}f_1(\vect x)
\amp=\frac{\partial}{\partial x_1}f_3(\vect x)\\
\frac{\partial}{\partial x_2}f_1(\vect x)
\amp=\frac{\partial}{\partial x_1}f_2(\vect x)\text{.}
\end{align*}
If we introduce a new differential expression on vector fields in \(\mathbb R^3\) we can state the above conditions in a very concise form.
Definition 8.14. Curl of a vector field.
If \(\vect f=(f_1,f_2,f_3)\) is a vector field defined on a subset of \(\mathbb R^3\) we set
\begin{equation}
\curl\vect f(\vect x)
:=\nabla\times\vect f(\vect x)
:=
\begin{bmatrix}
\dfrac{\partial}{\partial x_2}f_3(\vect x)
-\dfrac{\partial}{\partial x_3}f_2(\vect x) \\
\dfrac{\partial}{\partial x_3}f_1(\vect x)
-\dfrac{\partial}{\partial x_1}f_3(\vect x) \\
\dfrac{\partial}{\partial x_1}f_2(\vect x)
-\dfrac{\partial}{\partial x_2}f_1(\vect x)
\end{bmatrix}\text{.}\tag{8.6}
\end{equation}
Definition 8.15. The nabla operator.
The symbol \(\nabla\) is called the nabla operator, and can be thought of as the vector
\begin{equation*}
\nabla=\Bigl(\frac{\partial}{\partial x_1},
\frac{\partial}{\partial x_2},
\frac{\partial}{\partial x_3}\Bigr)
\end{equation*}
With the above notation the curl of
\(\vect f\) is formally the vector product of
\(\nabla\) with
\(\vect f\) as defined in
Definition 1.20. Hence the notation
\(\nabla\times\vect f\text{.}\) Moreover, the gradient of a scalar function,
\(f\text{,}\) is formally the multiplication by scalars of the `vector’
\(\nabla\) and the scalar
\(f\text{,}\) so we write
\(\nabla f\) for the gradient. In the more applied literature most of the time the `nabla operator’ is used.
With these definitions we have the following fact.
Fact 8.16.
A vector field, \(\vect f\text{,}\) defined on a subset of \(\mathbb R^3\) is closed if and only if \(\curl\vect f=0\) on its domain.
The curl will appear again later when discussing the Theorem of Stokes.
