Closed Vector Fields in Space

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Section 8.4 Closed Vector Fields in Space

If

N = 3

and and

f = (f_{1}, f_{2}, f_{3})

is a vector field then

f

is closed if

\begin{aligned} \frac{\partial}{\partial x_{2}} f_{3} (x) & = \frac{\partial}{\partial x_{3}} f_{2} (x) \\ \frac{\partial}{\partial x_{3}} f_{1} (x) & = \frac{\partial}{\partial x_{1}} f_{3} (x) \\ \frac{\partial}{\partial x_{2}} f_{1} (x) & = \frac{\partial}{\partial x_{1}} f_{2} (x) . \end{aligned}

If we introduce a new differential expression on vector fields in

R^{3}

we can state the above conditions in a very concise form.

Definition 8.14. Curl of a vector field.

If

f = (f_{1}, f_{2}, f_{3})

is a vector field defined on a subset of

R^{3}

we set

\begin{matrix} (8.6) & curl f (x) := \nabla \times f (x) := [\begin{matrix} \frac{\partial}{\partial x_{2}} f_{3} (x) - \frac{\partial}{\partial x_{3}} f_{2} (x) \\ \frac{\partial}{\partial x_{3}} f_{1} (x) - \frac{\partial}{\partial x_{1}} f_{3} (x) \\ \frac{\partial}{\partial x_{1}} f_{2} (x) - \frac{\partial}{\partial x_{2}} f_{1} (x) \end{matrix}] . \end{matrix}

Definition 8.15. The nabla operator.

The symbol

\nabla

is called the nabla operator, and can be thought of as the vector

\nabla = (\frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial x_{2}}, \frac{\partial}{\partial x_{3}})

With the above notation the curl of

f

is formally the vector product of

\nabla

with

f

as defined in Definition 1.20. Hence the notation

\nabla \times f .

Moreover, the gradient of a scalar function,

f,

is formally the multiplication by scalars of the `vector’

\nabla

and the scalar

f,

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,

f,

defined on a subset of

R^{3}

is closed if and only if

curl f = 0

on its domain.

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

f (x_{1}, x_{2}, x_{3}) := (f_{1} (x_{1}, x_{2}), f_{2} (x_{1}, x_{2}), 0)

in

R^{3} .

(The vectors

f (x)

do not depend on the third variable.) By definition of the curl we get

curl f (x) = [\begin{matrix} 0 \\ 0 \\ \frac{\partial}{\partial x_{1}} f_{2} (x) - \frac{\partial}{\partial x_{2}} f_{1} (x) \end{matrix}] .

Hence

curl f (x) = 0

if and only if (8.5) holds. Hence,

\frac{\partial}{\partial x_{1}} f_{2} (x) - \frac{\partial}{\partial x_{2}} f_{1} (x)

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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