Vector Fields

Section 6.1 Vector Fields

Revised on March 27.

References.

OpenStax Calculus Volume 3, Section 6.1
¹
openstax.org/books/calculus-volume-3/pages/6-1-vector-fields
.
Calculus, Early Transcendentals by Stewart, Section 16.1.

Subsection 6.1.1 Definitions

Many physical quantities are described by vectors (like velocity, force) and depend on position in space or on a surface. The mathematical description of these is a vector field:

Definition 6.1.1. 2D Vector Field.

For \(D\) a set in \(\mathbb{R}^2\) (a plane region), a vector field on \(\mathbb{R}^2\) is a function \(\vec{F}\) that assigns to each point \((x,y)\) in \(D\) a two-dimensional vector \(\vec{F}(x,y)\text{.}\)

Such a vector field can be described with component functions as \(\vec{F} = P\veci + Q\vecj\text{;}\) that is

\begin{equation*} \vec{F}(x,y) = P(x,y)\veci + Q(x,y)\vecj = \vector{P(x,y),Q(x,y)} \end{equation*}

Scalar valued functions of several variables like \(P\) and \(Q\) are sometimes called scalar fields to distinguish them from vector fields.

There is an unsurprising 3D version:

Definition 6.1.2. 3D Vector Field.

For \(E\) a set in \(\mathbb{R}^3\text{,}\) a vector field on \(\mathbb{R}^3\) is a function \(\vec{F}\) that assigns to each point \((x,y,z)\) in \(E\) a three-dimensional vector \(\vec{F}(x,y,z)\text{.}\)

Again such a vector field can be described with component functions, so

\begin{equation*} \vec{F}(x,y,z) = P(x,y,z)\veci + Q(x,y,z)\vecj + R(x,y,z)\vecj. \end{equation*}

Subsection 6.1.2 Gradient Vector Fields, or Conservative Vector Fields

Vector fields often arise as the gradient of a scalar function: for a function \(f\) of two variables the gradient

\begin{equation} \Grad f(x,y) = \vector{f_x,f_y} = f_x(x,y) \veci + f_y(x,y)\vecj\tag{6.1.1} \end{equation}

which is a vector field on \(\mathbb{R}^2\text{,}\) and likewise for \(f\) a function of three variables, its gradient is a vector field on \(\mathbb{R}^3\text{:}\)

\begin{equation} \Grad f(x,y,z) = \vector{f_x,f_y,f_z} = f_x(x,y,z)\veci + f_y(x,y,z)\vecj + f_z(x,y,z)\veck.\tag{6.1.2} \end{equation}

Two common short-hands for either of these are

\begin{equation} \Grad f = \nabla f = \del f\tag{6.1.3} \end{equation}

where the symbol \(\nabla\) or \(\del\) is the fake vector

\begin{equation} \nabla f = \del f = \vector{\partial_x, \partial_y}\tag{6.1.4} \end{equation}

in the plane, and

\begin{equation} \nabla f = \del f = \vector{\partial_x, \partial_y, \partial_z}\tag{6.1.5} \end{equation}

in space.

The second "vector" form of this notation will be more important later, from Section 6.5, Divergence and Curl onward.

Definition 6.1.3.

A vector field \(\vec{F}\) is called a gradient vector field or conservative vector field if it is the gradient of some scalar function; that is, if there is a function \(f\) such that \(\nabla f = \vec{F}\text{.}\)

Function \(f\) is then called a potential function for \(\vec{F}\text{.}\)

Subsection 6.1.3 The Cross-Partial Property and Non-Conservative Vector Fields

For a conservative 2D vector field \(\vec{F} = \vector{P, Q}, = \vector{f_x, f_y}\text{,}\) look at the cross partial derivatives:

\begin{equation*} P_y = (f_x)_y = (f_y)_x = Q_x \end{equation*}

(recalling Clairaut’s Theorem 4.3.2 from Section 4.3)

This can be extended to 3D vector fields, and can be written in a suggestive mock-determinant notation that we will see more of in Section 6.5:

Theorem 6.1.4. The Cross-Partial Property of Conservative Vector Fields.

For a 2D conservative vector field \(\vec{F} = \vector{P, Q}\text{,}\)

\begin{equation} Q_x - P_y = \left| \begin{array}{cc} \partial_x \amp \partial_y \\ P \amp Q \end{array} \right| = 0\tag{6.1.6} \end{equation}

For a 3D conservative vector field \(\vec{F} = \vector{P, Q, R}\text{,}\)

\begin{equation} Q_x - P_y = R_x - P_z = R_y - Q_z = 0,\tag{6.1.7} \end{equation}

which has the menomonic mock-determinant form

\begin{equation} \left| \begin{array}{ccc} \veci \amp \vecj \amp \veck \\ \partial_x \amp \partial_y \amp \partial_z \\ P \amp Q \amp R \end{array} \right|\tag{6.1.8} \end{equation}

Note: Here I start using the alternative compact notation for partial derivatives introduced in (4.3.3) and (4.3.4) in Subsection 4.3.3.

Exercise 6.1.5. A non-conservative vector field.

Verify that \(\vec{F}(x,y) = \vector{-y, x}\) is not conservative.

Exercise 6.1.6. A non-conservative vector field that has the cross-partial property.

Verify that the vector field

\begin{equation*} \ds\vec{F}(x,y) = \vector{-\frac{y}{x^2+y^2}, \frac{x}{x^2+y^2}} \end{equation*}

has the cross-partial property (6.1.6).

However, it will be seen in Section 6.3 that is not a gradient vector field on its natural domain of all points in \(\reals^2\) except for the hole at the origin.

Study Guide.

Study Section 6.1 of Calculus Volume 3

⁶

openstax.org/books/calculus-volume-3/pages/6-1-vector-fields

; in particular

The Definition and Theorems.
Examples 1–5 and 10–13, and the Checkpoints following each.
The following exercises (for ranges, do at least one in each range): 1, 2, 15–20, 22–24.

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