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Lessons for Multivariable Calculus

Mark Fitch

Section 5.5 Curl

Subsection 5.5.1 Definition

Definition 5.5.1. Curl.

For a 3D vector field \(\vec{F}\) the curl at a point is
\begin{equation*} \mbox{curl } \vec{F} = \left(\frac{\partial \vec{F_z}}{\partial y}-\frac{\partial \vec{F_y}}{\partial z},-\left(\frac{\partial \vec{F_z}}{\partial x}-\frac{\partial \vec{F_x}}{\partial z}\right), \frac{\partial \vec{F_y}}{\partial x}-\frac{\partial \vec{F_x}}{\partial y} \right). \end{equation*}

Subsection 5.5.2 Interpretation

Activity 75.

The purpose of this activity is to illustrate visually an interpretation of curl that explains the name.
Use Figure 5.5.2 to answer these questions. Note that the field is \(\vec{F}(x,y,z)=\langle \sin(\pi y), \cos(\pi x), 1 \rangle\text{.}\) The curl is given by \(\text{curl}(\vec{F})=\langle 0, 0, -\pi\sin(\pi x)-\pi \cos(\pi y) \rangle\text{.}\)
(a)
Would ball 1 spin clockwise, not at all or counterclockwise?
(b)
What is the curl at the center of ball 1 \((0.5,0,0)\text{?}\)
(c)
Would ball 2 spin clockwise, not at all or counterclockwise?
(d)
What is the curl at the center of ball 2 \((-0.5,1,0)\text{?}\)
(e)
Would ball 3 spin clockwise, not at all or counterclockwise?
(f)
What is the curl at the center of ball 3 \((-1.5,-1,0)\text{?}\)
Figure 5.5.2. Curl Illustrated by Balls in a Field

Subsection 5.5.3 Curl and Conservative Fields

Activity 76.

The purpose of this activity is to discover a connection between curl and conservative fields.
(b)
Compare Clairaut’s Theorem to the curl formula.
(c)
Review the criteria for a vector field to be conservative.
(d)
Write a new condition for a field to be conservative using these conclusions.

Exercises 5.5.4 Exercises

Calculate the curl at the indicated points. Determine which fields are conservative.

1.

\(\vec{F}(x,y,z)=\langle -x^2,-y^2,-z^2 \rangle\)
(a)
\(\mbox{curl}(\vec{F})(0,0,0)\)
(b)
\(\mbox{curl}(\vec{F})(-1,-1,-1)\)
(c)
\(\mbox{curl}(\vec{F})(1,1,1)\)

2.

\(\vec{G}(x,y,z)=\langle x^3,y^3,3 \rangle\)
(a)
\(\mbox{curl}(\vec{G})(0,0,0)\)
(b)
\(\mbox{curl}(\vec{G})(-1,-1,-1)\)
(c)
\(\mbox{curl}(\vec{G})(1,1,1)\)

3.

\(\vec{H}(x,y,z)=\langle x,xy,0 \rangle\)
(a)
\(\mbox{curl}(\vec{H})(0,0,0)\)
(b)
\(\mbox{curl}(\vec{H})(-2,-1,-1)\)
(c)
\(\mbox{curl}(\vec{H})(2,1,1)\)

4.

\(\vec{J}(x,y,z)=\langle z^3y,x^2z,xy \rangle\)
(a)
\(\mbox{curl}(\vec{J})(0,0,0)\)
(b)
\(\mbox{curl}(\vec{J})(-1,-1,-1)\)
(c)
\(\mbox{curl}(\vec{J})(1,1,1)\)
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