We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Curl

Study concepts, example questions & explanations for Calculus 3

Create An Account

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Curl

Calculate the curl for the following vector field.

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

Possible Answers:

$curl\ \vec{F}=\Big(4x^2y^3z^3\Big)\vec{i}+\Big(2xz^2\Big)\vec{j}+\Big(-2xy^3z^4+2x^3y\Big)\vec{k}$

$curl\ \vec{F}=0$

$curl\ \vec{F}=\Big(-4x^2y^3z^3\Big)\vec{i}+\Big(-2xz^2\Big)\vec{j}+\Big(2xy^3z^4-2x^3y\Big)\vec{k}$

$curl\ \vec{F}=\Big(2x^2z^3\Big)\vec{i}+\Big(2yz^2\Big)\vec{j}+\Big(-2x^3y\Big)\vec{k}$

Correct answer:

$curl\ \vec{F}=\Big(-4x^2y^3z^3\Big)\vec{i}+\Big(-2xz^2\Big)\vec{j}+\Big(2xy^3z^4-2x^3y\Big)\vec{k}$

Explanation:

In order to calculate the curl, we need to recall the formula.

$\\curl\ \vec{F}=\triangledown \times \vec{F}=\begin{vmatrix} \vec{i} & \vec{j} & \vec{k}\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}\\ \\ \\ =\Big(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\ \Big)\vec{i}+\Big(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\ \Big)\vec{j}+\Big(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\ \Big)\vec{k}$

where , , and correspond to the components of a given vector field: $\vec{F}=P\vec{i}+Q\vec{j}+R\vec{k}$

Now lets apply this to out situation.

$curl\ \vec{F}=\triangledown \times \vec{F}=\begin{vmatrix} \vec{i} & \vec{j} & \vec{k}\\\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\\\ x^3y^2 & x^2y^3z^4 & x^2z^2 \end{vmatrix}$

$\\=\Big(\frac{\partial }{\partial y}\Big(x^2z^2\Big)-\frac{\partial}{\partial z}\Big(x^2y^3z^4\Big)\ \Big)\vec{i}+\Big(\frac{\partial}{\partial z}\Big(x^3y^2\Big)-\frac{\partial}{\partial x}\Big(x^2z^2\Big)\ \Big)\vec{j}+\Big(\frac{\partial}{\partial x}\Big(x^2y^3z^4\Big)-\frac{\partial}{\partial y}\Big(x^3y^2\Big)\ \Big)\vec{k}$

$=\Big(0-4x^2y^3z^3\Big)\vec{i}+\Big(0-2xz^2\Big)\vec{j}+\Big(2xy^3z^4-2x^3y\Big)\vec{k}$

Thus the curl is

$=\Big(-4x^2y^3z^3\Big)\vec{i}+\Big(-2xz^2\Big)\vec{j}+\Big(2xy^3z^4-2x^3y\Big)\vec{k}$

Report an Error

Example Question #2 : Curl

Calculate the curl for the following vector field.

$\vec{F}=xz\ \vec{i}+x^2yz\ \vec{j}+y^2z^2\ \vec{k}$

Possible Answers:

$curl \ \vec{F}=\Big(2yz^2-x^2y\Big)\vec{i}+\Big(x\Big)\vec{j}+\Big(2xyz\Big)\vec{k}$

$curl \ \vec{F}=\Big(-x^2y\Big)\vec{i}+\Big(-x\Big)\vec{j}+\Big(xyz\Big)\vec{k}$

$curl \ \vec{F}=\Big(2yz^2\Big)\vec{i}+\Big(z\Big)\vec{j}+\Big(2xz\Big)\vec{k}$

$curl \ \vec{F}=\Big(-2yz^2+x^2y\Big)\vec{i}+\Big(-x\Big)\vec{j}+\Big(-2xyz\Big)\vec{k}$

$curl \ \vec{F}=0$

Correct answer:

$curl \ \vec{F}=\Big(2yz^2-x^2y\Big)\vec{i}+\Big(x\Big)\vec{j}+\Big(2xyz\Big)\vec{k}$

Explanation:

In order to calculate the curl, we need to recall the formula.

where , , and correspond to the components of a given vector field: $\vec{F}=P\vec{i}+Q\vec{j}+R\vec{k}$

Now lets apply this to out situation.

$\\=\Big(\frac{\partial }{\partial y}\Big(y^2z^2\Big)-\frac{\partial}{\partial z}\Big(x^2yz\Big)\ \Big)\vec{i}+\Big(\frac{\partial}{\partial z}\Big(xz\Big)-\frac{\partial}{\partial x}\Big(y^2z^2\Big)\ \Big)\vec{j}+\Big(\frac{\partial}{\partial x}\Big(x^2yz\Big)-\frac{\partial}{\partial y}\Big(xz\Big)\ \Big)\vec{k}$

$=\Big(2yz^2-x^2y\Big)\vec{i}+\Big(x-0\Big)\vec{j}+\Big(2xyz-0\Big)\vec{k}$

Thus the curl is

$=\Big(2yz^2-x^2y\Big)\vec{i}+\Big(x\Big)\vec{j}+\Big(2xyz\Big)\vec{k}$

Report an Error

Example Question #3 : Curl

Find the curl of the force field $\mathbf{F}(x,y,z) = \mathbf{i} +xy\mathbf{j} +z^2\mathbf{k}$

Possible Answers:

None of the other answers

All of the other answers

$\mathbf{i} +xy\mathbf{j} +z^2\mathbf{k}$

1+xy+z^2

$y\mathbf{k}$

Correct answer:

$y\mathbf{k}$

Explanation:

Curl is probably best remembered by the determinant formula

$\bigtriangledown \times \mathbf{F}(x,y,z)$ , which is used here as follows.

$\bigtriangledown \times \mathbf{F}(x,y,z) = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} &\frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 1& xy & z^2 \end{vmatrix}$

$=\begin{vmatrix} \frac{\partial}{\partial y} &\frac{\partial}{\partial z} \\ xy & z^2 \end{vmatrix}\mathbf{i}- \begin{vmatrix} \frac{\partial}{\partial x} &\frac{\partial}{\partial z} \\ 1 & z^2 \end{vmatrix} \mathbf{j}+ \begin{vmatrix} \frac{\partial}{\partial x} &\frac{\partial}{\partial y} \\ 1 & xy \end{vmatrix}\mathbf{k}$

$=(\frac{\partial}{\partial y}z^2 -\frac{\partial}{\partial z}(xy))\mathbf{i} -(\frac{\partial}{\partial x}z^2 - \frac{\partial}{\partial z}(1))\mathbf{j} + (\frac{\partial}{\partial x}(xy)-\frac{\partial}{\partial y}(1))\mathbf{k}$ $=0\mathbf{i} - 0\mathbf{j}+ (y-0)\mathbf{k}$

$=y\mathbf{k}$

Report an Error

Example Question #1 : Curl

Let be any arbitrary real valued vector field. Find the div(curl F)

Possible Answers:

$\pm\frac{1}{2}$

Correct answer:

Explanation:

Take any field, the curl gives us the amount of rotation in the vector field. The purpose of the divergence is to tell us how much the vectors move in a linear motion.

When vectors are moving in circular motion only, there are no possible linear motion. Thus the divergence of the curl of any arbitrary vector field is zero.

Report an Error

Example Question #1 : Curl

Evaluate the curl of the force field $F = (x^3+1)\mathbf{i}+xy\mathbf{j}+\ln(z)\mathbf{k}$ .

Possible Answers:

$\mathbf{0}$

$z\mathbf{k}$

$y\mathbf{i}$

$y\mathbf{k}$

$x\mathbf{i}$

Correct answer:

$y\mathbf{k}$

Explanation:

To evaluate the curl of a force field, we use Curl $(F)=\bigtriangledown \times F.$

$\bigtriangledown \times F = \begin{vmatrix} \mathbf{i}&\mathbf{j} &\mathbf{k} \\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y} &\frac{\partial}{\partial z} \\ (x^3 +1)& xy & \ln(z) \end{vmatrix}$ . Start

$=(\frac{\partial}{\partial y} \ln(z) - \frac{\partial}{\partial z}(xy))\mathbf{i} -(\frac{\partial}{\partial x}\ln(z)-\frac{\partial}{\partial z}(x^3+1))\mathbf{j} +(\frac{\partial}{\partial x}(xy)-\frac{\partial}{\partial y}(x^3+1))\mathbf{k}$ Evaluate along the first row using cofactor expansion.

$= y\mathbf{k}$ . Evaluate partial derivatives. All terms except the 2nd to last one are .

Report an Error

Example Question #1 : Curl

Calculate the curl $\triangledown \times \mathbf{F}$ of the following vector:

$\mathbf{F}=xz\mathbf{i}+xy\mathbf{j}+yz\mathbf{k}$

Possible Answers:

$z\mathbf{i}+x\mathbf{j}-y\mathbf{k}$

$-z\mathbf{i}+x\mathbf{j}-y\mathbf{k}$

$z\mathbf{i}+x\mathbf{j}+y\mathbf{k}$

$z\mathbf{i}-x\mathbf{j}+y\mathbf{k}$

$-z\mathbf{i}-x\mathbf{j}-y\mathbf{k}$

Correct answer:

$z\mathbf{i}+x\mathbf{j}+y\mathbf{k}$

Explanation:

The curl of a vector $\mathbf{F}=U\mathbf{i}+V\mathbf{j}+W\mathbf{k}$

is defined by the determinant of the following 3x3 matrix:

$\triangledown \times \mathbf{F}=\textup{det} \begin{vmatrix} \mathbf{i}& \mathbf{j} &\mathbf{k} \\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y} &\frac{\partial}{\partial z} \\ U& V& W \end{vmatrix}$

For the given vector, we can calculate this determinant

$\triangledown \times \mathbf{F}=(z-0)\mathbf{i} -(0-x)\mathbf{j}+(y-0)\mathbf{k}=z\mathbf{i}+x\mathbf{j}+y\mathbf{k}$

Report an Error

Example Question #7 : Curl

Given that F is a vector function and f is a scalar function, which of the following operations results in a scalar?

Possible Answers:

$\nabla \cdot(\nabla \times \mathbf{F})$

$\nabla \cdot(\nabla \cdot\mathbf{F})$

$\nabla \times(\nabla f)$

$\nabla(\nabla \cdot\mathbf{F})$

$\nabla(\nabla\cdot f)$

Correct answer:

$\nabla \cdot(\nabla \times \mathbf{F})$

Explanation:

For each of the given expressions:

$\nabla(\nabla \cdot f)$ - The divergence of a scalar function does not exist, so this expression is undefined.

$\nabla(\nabla \cdot\mathbf{F})$ - The dot product of a vector function is a scalar, so the gradient of the term in parenthesis results in a vector.

$\nabla \cdot(\nabla \cdot\mathbf{F})$ - The divergence of a vector function is a scalar. Taking the divergence of the term in parenthesis would be taking the divergence of a scalar, which doesn't exist. This expression is undefined.

$\nabla \times(\nabla f)$ - The gradient of a scalar function is a vector. Thus, the curl of the term in parenthesis is also a vector.

The remaining answer is:

$\nabla \cdot(\nabla \times \mathbf{F})$ - The term in parenthesis is the curl of a vector function, which is also a vector. Taking the divergence of the term in parenthesis, we get the divergence of a vector, which is a scalar.

Report an Error

Example Question #1 : Curl

Given that F is a vector function and f is a scalar function, which of the following expressions is undefined?

Possible Answers:

$\nabla \times (\nabla f)$

$\nabla\cdot(\nabla f)$

$\nabla(\nabla\cdot\mathbf{F})$

$\nabla \times (\nabla\times f)$

$\nabla\cdot(\nabla\times\mathbf{F})$

Correct answer:

$\nabla \times (\nabla\times f)$

Explanation:

The cross product of a scalar function is undefined. The expression in the parenthesis of:

$\nabla \times (\nabla\times f)$

is the cross product of a scalar function, therefore the entire expression is undefined.

For the other solutions:

$\nabla\cdot(\nabla\times\mathbf{F})$ - The cross product of a vector is also a vector, and the divergence of a vector is defined. This expression is a scalar.

$\nabla\cdot(\nabla f)$ - The gradient of a scalar is a vector, and the divergence of a vector is defined. This expression is also a scalar.

$\nabla(\nabla\cdot\mathbf{F})$ - The divergence of a vector is scalar, and the gradient of a scalar is defined. This expression is a vector.

$\nabla \times (\nabla f)$ - The gradient of a scalar is a vector, and the curl of a vector is defined. This expression is a vector.

Report an Error

Example Question #1 : Curl

Compute the curl of the following vector function:

$\mathbf{F}=-y^2\mathbf{i}+3y\mathbf{j}$

Possible Answers:

$-2y\mathbf{i}+3\mathbf{k}$

$2y\mathbf{k}$

$3\mathbf{i}-2y\mathbf{k}$

$-2y\mathbf{i}$

$-3\mathbf{k}$

Correct answer:

$2y\mathbf{k}$

Explanation:

For a vector function $\mathbf{F}=U\mathbf{i}+V\mathbf{j}+W\mathbf{k}$ , the curl is given by:

$\nabla \times \mathbf{F}=\textup{det}\begin{vmatrix} \mathbf{i}&\mathbf{j} &\mathbf{k} \\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y} &\frac{\partial}{\partial z} \\ U& V& W \end{vmatrix}$

For this function, we calculate the curl as:

$\nabla \times \mathbf{F}=(0-0)\mathbf{i}-(0-0)\mathbf{j}+(0+2y)\mathbf{k}=2y\mathbf{k}$

Report an Error

Example Question #10 : Curl

Compute the curl of the following vector function:

$\mathbf{F}=(y^2+z^2)\mathbf{i}+y\mathbf{j}+z\mathbf{k}$

Possible Answers:

$2y\mathbf{i}+\mathbf{j}$

$(2y-1)\mathbf{i}-\mathbf{k}$

$(1-2z)\mathbf{i}+\mathbf{j}$

$2z\mathbf{j}-2y\mathbf{k}$

$-2z\mathbf{i}-\mathbf{k}$

Correct answer:

$2z\mathbf{j}-2y\mathbf{k}$

Explanation:

For a vector function $\mathbf{F}=U\mathbf{i}+V\mathbf{j}+W\mathbf{k}$ , the curl is given by:

For this function, we calculate the curl as:

$\nabla \times \mathbf{F} =\textup{det}\begin{vmatrix} \mathbf{i}&\mathbf{j} &\mathbf{k} \\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y} &\frac{\partial}{\partial z} \\ (y^2+z^2)& y& z \end{vmatrix}$

$\nabla\times \mathbf{F}=(0-0)\mathbf{i}-(0-2z)\mathbf{j}+(0-2y)\mathbf{k}= 2z\mathbf{j}-2y\mathbf{k}$

Report an Error

View Calculus 3 Tutors

Dossevi
Certified Tutor

cole Ste Genevive Versailles, Bachelor of Science, Mathematics. Institut National Polytechnique de Grenoble, Master of Scienc...

View Calculus 3 Tutors

Pavan
Certified Tutor

University of Tennessee, Knoxville, Master's/Graduate, Mechanical Engineering. Pennsylvania State University, Master's/Gradua...

View Calculus 3 Tutors

Asad
Certified Tutor

NED University of Engineering and Technology, Bachelor of Science, Electrical Engineering. University of Cincinnati-Main Camp...

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Biology Tutors in Seattle, Physics Tutors in Phoenix, MCAT Tutors in Miami, GRE Tutors in San Diego, French Tutors in New York City, GMAT Tutors in Atlanta, French Tutors in Dallas Fort Worth, Calculus Tutors in Chicago, Math Tutors in Atlanta, Spanish Tutors in Denver

Popular Courses & Classes

GMAT Courses & Classes in Philadelphia, ACT Courses & Classes in Houston, MCAT Courses & Classes in Washington DC, GMAT Courses & Classes in San Francisco-Bay Area, SAT Courses & Classes in Denver, GRE Courses & Classes in Seattle, SAT Courses & Classes in Houston, Spanish Courses & Classes in San Diego, Spanish Courses & Classes in Houston, LSAT Courses & Classes in Washington DC

Popular Test Prep

LSAT Test Prep in San Diego, SAT Test Prep in Atlanta, ACT Test Prep in Houston, GRE Test Prep in Atlanta, MCAT Test Prep in Seattle, GRE Test Prep in Seattle, SAT Test Prep in Seattle, SSAT Test Prep in Dallas Fort Worth, GRE Test Prep in Philadelphia, MCAT Test Prep in Miami

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Calculus 3 : Curl

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #1 : Curl

Example Question #2 : Curl

Example Question #3 : Curl

Example Question #1 : Curl

Example Question #1 : Curl

Example Question #1 : Curl

Example Question #7 : Curl

Example Question #1 : Curl

Example Question #1 : Curl

Example Question #10 : Curl

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: