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 vectors vector fields

### Calculus Topics Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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17calculus > vector fields > curl

 What is The Curl of a Vector? Calculating Curl Interpretation and Details Big Picture Curl in Cylindrical Coordinates Practice

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

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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