\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{\mathrm{sec} } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{\mathrm{arccot} } \) \( \newcommand{\arcsec}{\mathrm{arcsec} } \) \( \newcommand{\arccsc}{\mathrm{arccsc} } \) \( \newcommand{\sech}{\mathrm{sech} } \) \( \newcommand{\csch}{\mathrm{csch} } \) \( \newcommand{\arcsinh}{\mathrm{arcsinh} } \) \( \newcommand{\arccosh}{\mathrm{arccosh} } \) \( \newcommand{\arctanh}{\mathrm{arctanh} } \) \( \newcommand{\arccoth}{\mathrm{arccoth} } \) \( \newcommand{\arcsech}{\mathrm{arcsech} } \) \( \newcommand{\arccsch}{\mathrm{arccsch} } \)

17Calculus - Curl of Vector Fields

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The curl of a vector field gives an indication of how a vector field tends to curve or rotate. We use the idea of the cross product to calculate the curl. Before we get started on the details of the curl, let's watch a video to get started.

Michel vanBiezen - What is the Curl of a Vector? [7mins-12secs]

video by Michel vanBiezen

Calculating Curl

To calculate the curl of a vector field \( \vec{F}(x,y,z) = F_i\vhat{i} + F_j\vhat{j} + F_k\vhat{k}\), we use the del operator, \(\displaystyle{ \vec{ \nabla } = \frac{\partial }{\partial x}\vhat{i} + \frac{\partial }{ \partial y}\vhat{j} + \frac{\partial }{ \partial z}\vhat{k} }\) and the cross product to give us
\(\displaystyle{ \vec{ \nabla } \times \vec{F} = }\) \(\displaystyle{ \left[ \frac{\partial }{\partial x}\vhat{i} + \right. }\) \(\displaystyle{ \frac{\partial }{ \partial y}\vhat{j} + }\) \(\displaystyle{ \left. \frac{\partial }{ \partial z}\vhat{k} \right] \times \left[ F_i\vhat{i} + F_j\vhat{j} + F_k\vhat{k} \right] = }\) \( \begin{vmatrix} \vhat{i} & \vhat{j} & \vhat{k} \\ \displaystyle{\frac{\partial }{\partial x}} & \displaystyle{\frac{\partial }{\partial y}} & \displaystyle{\frac{\partial }{\partial z}} \\ F_i & F_j & F_k \end{vmatrix} \)

Notes
1. You should know enough about linear algebra to be able to evaluate the determinant.
2. Although we call the curl a cross product and use the idea of the determinant, neither of these terms are exactly correct since del is an operator, not a true vector. However, for our purposes we follow the lead of most mathematicians and stretch the notation a bit to fit our requirements.
3. The curl is also written \( curl ~ \vec{F} \), i.e. \( curl ~ \vec{F} = \vec{ \nabla } \times \vec{F} \).
4. The curl of two vector fields is another vector field, which is perpendicular to both of the original vector fields.

Okay, time for a video. Here is a quick, no-frills introduction to calculating the curl of vector fields. It contains some detail when evaluating the determinant, which may be helpful to you if you need a reminder on how to do that.

Dr Chris Tisdell - Curl of a vector field [5mins-6secs]

video by Dr Chris Tisdell

Interpretation and Details

This video clip explains the interpretation and properties of the curl, as well as the circulation density and the scalar curl. This is important to watch to get a full understanding of the curl.

Dr Chris Tisdell - What is the curl? [28mins-21secs]

video by Dr Chris Tisdell

Big Picture

The curl measures rotation in a vector field.

If the scalar curl is positive everywhere in the plane, then there is a counter-clockwise rotation in the plane.

If the scalar curl is negative everywhere in the plane, then there is a clockwise rotation in the plane.

A vector field with zero curl is called irrotational and is considered a conservative vector field.

This next video clip gives the big picture of the curl of a vector field. The list above is adapted from the notes of Dr Chris Tisdell. In this video clip, he discusses these items. [We highly recommend that you go to his website and download the notes for this topic.]

Dr Chris Tisdell - Curl of vector fields [1min-11secs]

video by Dr Chris Tisdell

Curl in Cylindrical Coordinates

As you would expect, you can calculate the curl of a vector field in cylindrical coordinates. Here is a video explaining the equations and how to use them.

Michel vanBiezen - Curl of a Cylindrical Vector Field [5mins-12secs]

video by Michel vanBiezen

If you are following so far, it is time for some practice problems. Once you are finished with those, your next topic is divergence of vector fields. Divergence is similar to the curl but measures something different. The curl measures rotation around a point. The divergence measures the dispersion away from or contraction toward a point.

Practice

Instructions - Unless otherwise instructed, calculate the curl of these vector fields. If a point is given, also calculate the curl at that point.
If the curl turns out to be zero and the solution shows calculation of the potential function, you can ignore that part of the solution, if you have not studied that yet.

Basic Problems

\( \vec{F}(x,y,z) = \langle x^2z, -2xz, yz \rangle\); \( (6,-3,1) \)

Problem Statement

Calculate the curl of the vector field \( \vec{F}(x,y,z) = \langle x^2z, -2xz, yz \rangle\); \( (6,-3,1) \)

Solution

842 video

video by MIP4U

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\( \vec{G}(x,y) = \langle y,0 \rangle \)

Problem Statement

Calculate the curl of the vector field \( \vec{G}(x,y) = \langle y,0 \rangle \)

Solution

844 video

video by Dr Chris Tisdell

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\( \vec{F} = (x^2-y)\hat{i} + 4x\hat{j} + x^2\hat{k} \)

Problem Statement

Calculate the curl of the vector field \( \vec{F} = (x^2-y)\hat{i} + 4x\hat{j} + x^2\hat{k} \)

Solution

845 video

video by Dr Chris Tisdell

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\( \vec{F}(x,y,z) = y^3\hat{i} + x^2\hat{j} \)

Problem Statement

Calculate the curl of the vector field \( \vec{F}(x,y,z) = y^3\hat{i} + x^2\hat{j} \)

Solution

846 video

video by Dr Chris Tisdell

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\( \vec{F} = x^2\hat{i} + z^3\hat{j} + x^2y^4\hat{k} \)

Problem Statement

Calculate the curl of the vector field \( \vec{F} = x^2\hat{i} + z^3\hat{j} + x^2y^4\hat{k} \)

Solution

847 video

video by Dr Chris Tisdell

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\( \vec{F} = y^2\hat{i}+x^3\hat{j} \)

Problem Statement

Calculate the curl of the vector field \( \vec{F} = y^2\hat{i}+x^3\hat{j} \)

Solution

848 video

video by Dr Chris Tisdell

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Intermediate Problems

\( \vec{F} = (1+z^2)\vhat{i} + xy\vhat{j} + x^2y\vhat{k} \)

Problem Statement

Calculate the curl of the vector field \( \vec{F} = (1+z^2)\vhat{i} + xy\vhat{j} + x^2y\vhat{k} \)

Solution

839 video

video by Dr Chris Tisdell

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\( \vec{F} = xyz\vhat{i} + \cos(xyz)\vhat{j} + xy^2z^3\vhat{k} \)

Problem Statement

Calculate the curl of the vector field \( \vec{F} = xyz\vhat{i} + \cos(xyz)\vhat{j} + xy^2z^3\vhat{k} \)

Solution

840 video

video by Dr Chris Tisdell

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\( \vec{F}(x,y,z) = \langle x^2-y, y+z, z^2-x \rangle \)

Problem Statement

Calculate the curl of the vector field \( \vec{F}(x,y,z) = \langle x^2-y, y+z, z^2-x \rangle \)

Solution

843 video

video by Dr Chris Tisdell

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