\( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \)
Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Radius of Convergence
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Arc Length
Surface Area
Polar Coordinates
Slope & Tangent Lines
Arc Length
Surface Area
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Ready For Calculus 2?
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Practice Exams
17calculus on YouTube
More Math Help
Tools and Resources
Academic Integrity
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Instructor or Coach?
Math Books
How To Read Math Books

You CAN Ace Calculus

17calculus > vector fields > curl

Curl of Vector Fields

on this page: ► calculating curl     ► interpretation and details     ► big picture     ► next

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? [7min-2secs]

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
\(\displaystyle{ \vec{ \nabla } \times \vec{F} = }\) \(\displaystyle{ \left[ \frac{\partial }{\partial x}\vhat{i} + \right. }\) \(\displaystyle{ \frac{\partial }{ \partial y}\vhat{j} + }\) \(\displaystyle{ \left. \frac{\partial }{ \partial z}\vhat{k} \right] \times \left[ F_i\vhat{i} + F_j\vhat{j} + F_k\vhat{k} \right] = }\) \(\displaystyle{ \begin{vmatrix} \vhat{i} & \vhat{j} & \vhat{k} \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ F_i & F_j & F_k \end{vmatrix} }\)

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

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?

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

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.

next: divergence →

Search 17Calculus

Practice Problems

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.

Level A - Basic

Practice A01

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


Practice A02

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


Practice A03

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


Practice A04

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


Practice A05

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


Practice A06

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


Practice A07

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


Level B - Intermediate

Practice B01

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


Practice B02

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


Practice B03

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


Real Time Web Analytics
menu top search practice problems
menu top search practice problems 17