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Calculus 3 (Math 221) Notes and Study Guide

Brenton LeMesurier

Section 6.1 Vector Fields

References.

Topics.

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)\veck. \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} \del f(x,y) = \vector{\partial_x f, \partial_y f} = \vector{f_x,f_y}\tag{6.1.1} \end{equation}
introduced in Subsection 4.6.2, The Gradient Vector 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} \del f(x,y,z) = \vector{f_x,f_y,f_z}\tag{6.1.2} \end{equation}

Remark 6.1.3.

Here we use the alternative notation “\(\partial_x\)” etc. for partial derivatives, which was introduced in (4.3.3) and (4.3.4).
In addition to the notations
\begin{equation} \Grad f = \nabla f = \del f\tag{6.1.3} \end{equation}
seen in Subsection 4.6.2, the symbol \(\nabla\) or \(\del\) is sometimes treated as the fake vector
\begin{equation} \del = \vector{\partial_x, \partial_y}\tag{6.1.4} \end{equation}
in the plane, and
\begin{equation} \del = \vector{\partial_x, \partial_y, \partial_z}\tag{6.1.5} \end{equation}
in space, which is “consistent” if you treat juxtaposition a bit like multiplication:
\begin{equation*} \del f = \vector{\partial_x, \partial_y} f = \vector{\partial_x f, \partial_y f} \end{equation*}
and so on.
This “vector” notation will be useful later, from Section 6.5, Divergence and Curl onward.

Definition 6.1.4.

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 \(\del f = \vec{F}\text{.}\)
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, \text{ so } Q_x - P_y = \partial_x Q - \partial_y P = 0 \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:

Exercise 6.1.6. A non-conservative vector field.

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

Exercise 6.1.7. 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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