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Vector Calculus: An Introduction

Daniel Daners

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}

Remark 8.15.

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*}
Then the curl of \(\vect f\) is formally the vector product of \(\nabla\) with \(\vect f\) as defined in Definition 1.21. 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.
The curl will appear again later when discussing the Theorem of Stokes.

Remark 8.17.

Let us make a remark about the relationship of the curl with the condition (8.5) for a plane vector field to be closed. If \((f_1,f_2)\) is a plane vector field we can always define a vector field \(\vect f(x_1,x_2,x_3):=(f_1(x_1,x_2),f_2(x_1,x_2),0)\) in \(\mathbb R^3\text{.}\) (The vectors \(\vect f(\vect x)\) do not depend on the third variable.) By definition of the curl we get
\begin{equation*} \curl\vect f(\vect x)= \begin{bmatrix} 0 \\ 0\\ \dfrac{\partial}{\partial x_1}f_2(\vect x) -\dfrac{\partial}{\partial x_2}f_1(\vect x) \end{bmatrix}\text{.} \end{equation*}
Hence \(\curl \vect f(\vect x)=0\) if and only if (8.5) holds. Hence,
\begin{equation*} \frac{\partial}{\partial x_1}f_2(\vect x) -\frac{\partial}{\partial x_2}f_1(\vect x) \end{equation*}
is often called the curl of a plane vector field. Using the Theorem of Stokes we will give a physical interpretation of the curl. For details see Observation 11.8 below.
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