Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Calculus 3

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 #1741 : Calculus 3

Determine the curl of the following vector field:

$\vec{F}=6x\hat{i}+(e^z+y)\hat{j}+z\hat{k}$

Possible Answers:

$-e^z\hat{i}$

$-(e^z+y)\hat{i}$

$e^z\hat{i}$

$\vec{0}$

Correct answer:

$-e^z\hat{i}$

Explanation:

The curl of the vector field is given by

$\begin{vmatrix} \hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ 6x&e^z+y & z \end{vmatrix}$

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

Next, we take the cross product. One can do this by multiplying across from the top left to the lower right, and continuing downward, and then subtracting the terms multiplied from top right to the bottom left. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

$\frac{\partial }{\partial y}z\hat{i}+\frac{\partial }{\partial x}(e^z+j)\hat{k}+\frac{\partial }{\partial z}6x\hat{j}-(\frac{\partial }{\partial y}6x\hat{k}+\frac{\partial }{\partial z}(e^z+j)\hat{i}+\frac{\partial }{\partial x}z\hat{j})=-e^z\hat{i}$

Report an Error

Example Question #1742 : Calculus 3

Determine the curl of the following vector field:

$\vec{F}=\ln(y)\hat{i}+xz^2\hat{j}+y\hat{k}$

Possible Answers:

$(1-2xz)\hat{i}+(z^2-\frac{1}{y})\hat{k}$

$(1+2xz)\hat{i}+(z^2+\frac{1}{y})\hat{k}$

$(1-xz^2)\hat{i}+(z^2-\frac{1}{y})\hat{k}$

$(1-2xz)\hat{i}+(z^2-\ln(y))\hat{k}$

Correct answer:

$(1-2xz)\hat{i}+(z^2-\frac{1}{y})\hat{k}$

Explanation:

The curl of the vector field is given by

$\begin{vmatrix} \hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ \ln(y)&xz^2 & y \end{vmatrix}$

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

$\frac{\partial }{\partial y}y\hat{i}+\frac{\partial }{\partial x}xz^2\hat{k}+\frac{\partial }{\partial z}\ln(y)\hat{j}-(\frac{\partial }{\partial y}\ln(y)\hat{k}+\frac{\partial }{\partial z}xz^2\hat{i}+\frac{\partial }{\partial x}y\hat{j})=(1-2xz)\hat{i}+(z^2-\frac{1}{y})\hat{k}$

Report an Error

Example Question #20 : Curl

Find the curl of the following vector field:

$\vec{F}=\left \langle xy^2, x^2, \cos(z) \right \rangle$

Possible Answers:

$(2x-2xy)\hat{j}$

$(2x+2xy)\hat{k}$

2x-2xy

$(2x-2xy)\hat{k}$

$\vec{0}$

Correct answer:

$(2x-2xy)\hat{k}$

Explanation:

The curl of the vector field is given by

$\begin{vmatrix} \hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ xy^2&x^2 & \cos(z) \end{vmatrix}$

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

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

Report an Error

Example Question #21 : Curl

Find the curl of the vector field:

$\vec{F}=\sec(y)\hat{i}+xy\hat{j}+y^2\hat{k}$

Possible Answers:

$\left \langle 2y, 0, y-\sec(y)\tan(y)\right \rangle$

$\left \langle y, 0, y-\sec(y)\tan(y)\right \rangle$

$\left \langle 2y, 0, y+\sec(y)\tan(y)\right \rangle$

$\left \langle 2y, 0, y-\sec(y)\tan(y)\right \rangle$

Correct answer:

$\left \langle 2y, 0, y-\sec(y)\tan(y)\right \rangle$

Explanation:

The curl of the vector field is given by

$\begin{vmatrix} \hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ \sec(y)&xy &y^2 \end{vmatrix}$

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

$\frac{\partial }{\partial y}y^2\hat{i}+\frac{\partial }{\partial x}xy\hat{k}+\frac{\partial }{\partial z}\sec(y)\hat{j}-(\frac{\partial }{\partial y}\sec(y)\hat{k}+\frac{\partial }{\partial z}xy\hat{i}+\frac{\partial }{\partial x}y^2\hat{j})=2y\hat{i}+(y-\sec(y)\tan(y))\hat{k}=\left \langle 2y, 0, y-\sec(y)\tan(y)\right \rangle$

Report an Error

Example Question #91 : Line Integrals

Find the curl of the vector field:

$\vec{F}=\left \langle y^2, \sin(x), z\right \rangle$

Possible Answers:

$\left \langle 0, \cos(x)-2y, 0\right \rangle$

$\left \langle 0, 0, \cos(x)-2y\right \rangle$

$\left \langle 0, 0, \cos(x)+2y\right \rangle$

$\left \langle 0, 0, -\cos(x)+2y\right \rangle$

Correct answer:

$\left \langle 0, 0, \cos(x)-2y\right \rangle$

Explanation:

The curl of the vector field is given by

$\begin{vmatrix} \hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ y^2&\sin(x) &z \end{vmatrix}$

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

$\frac{\partial }{\partial y}z\hat{i}+\frac{\partial }{\partial x}\sin(x)\hat{k}+\frac{\partial }{\partial z}y^2\hat{j}-(\frac{\partial }{\partial y}y^2\hat{k}+\frac{\partial }{\partial z}\sin(x)\hat{i}+\frac{\partial }{\partial x}z\hat{j})=(\cos(x)-2y)\hat{k}=\left \langle 0, 0, \cos(x)-2y\right \rangle$

Report an Error

Example Question #21 : Curl

Determine whether the following vector field is conservative or not, and why:

$\vec{F}=\left \langle x\cos(y), xz, y^2\right \rangle$

Possible Answers:

The vector field is conservative because the curl equals zero.

The vector field is conservative because the curl does not equal zero.

The vector field is not conservative because the curl equals zero.

The vector field is not conservative because the curl does not equal zero.

Correct answer:

The vector field is not conservative because the curl does not equal zero.

Explanation:

The curl of the vector field is given by

$\begin{vmatrix} \hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ x\cos(y)&xz &y^2 \end{vmatrix}$

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

$\frac{\partial }{\partial y}y^2\hat{i}+\frac{\partial }{\partial x}xz\hat{k}+\frac{\partial }{\partial z}x\cos(y)\hat{j}-(\frac{\partial }{\partial y}x\cos(y)\hat{k}+\frac{\partial }{\partial z}xz\hat{i}+\frac{\partial }{\partial x}y^2\hat{j})=2y\hat{i}+(z+x\sin(y))\hat{k}\neq \vec{0}$

So, the vector field is not conservative because the curl did not result in the zero vector.

Report an Error

Example Question #24 : Curl

Determine whether the vector field is conservative or not, and why:

$\vec{F}=\left \langle x^2z, xy\cos(y), y^2\right \rangle$

Possible Answers:

The vector field is conservative because the curl is zero.

The vector field is not conservative because the curl is zero.

The vector field is conservative because the curl is not zero.

The vector field is not conservative because the curl is not zero.

Correct answer:

The vector field is not conservative because the curl is not zero.

Explanation:

A vector field is conservative if its curl is equal to zero (the zero vector).

The curl of the vector field is given by

$\begin{vmatrix} \hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ x^2z& xy\cos(y)& y^2 \end{vmatrix}$

$\frac{\partial }{\partial y}y^2\hat{i}+\frac{\partial }{\partial x}xy\cos(y)\hat{k}+\frac{\partial }{\partial z}x^2z\hat{j}-(\frac{\partial }{\partial y}x^2z\hat{k}+\frac{\partial }{\partial z}xy\cos(y)\hat{i}+\frac{\partial }{\partial x}y^2\hat{j})= 2y \hat{i}+y\cos(y)\hat{k}+x^2\hat{j}= \left \langle 2y, y\cos(y), x^2\right \rangle\neq \vec{0}$

The curl is not equal to the zero vector, so the vector field is not conservative.

Report an Error

Example Question #22 : Curl

Determine whether the vector field is conservative or not, and why:

$\vec{F}=\left \langle x, y^4e^y, 3z^2\right \rangle$

Possible Answers:

The vector field is not conservative because the curl is zero.

The vector field is conservative because the curl is not zero.

The vector field is conservative because the curl is zero.

The vector field is not conservative because the curl is not zero.

Correct answer:

The vector field is conservative because the curl is zero.

Explanation:

A vector field is conservative if its curl is equal to zero (the zero vector).

The curl of the vector field is given by

$\begin{vmatrix} \hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ x& y^4e^y\cos(y)& 3z^2 \end{vmatrix}$

$\frac{\partial }{\partial y}3z^2\hat{i}+\frac{\partial }{\partial x}y^4e^y\hat{k}+\frac{\partial }{\partial z}x\hat{j}-(\frac{\partial }{\partial y}x\hat{k}+\frac{\partial }{\partial z}y^4e^y\hat{i}+\frac{\partial }{\partial x}3z^2\hat{j})= \vec{0}$

All of the partial derivatives returned zero values for the unit vectors, so the curl indeed is equal to zero. The vector field is conservative.

Report an Error

Example Question #1743 : Calculus 3

Find the curl of the vector field:

$\vec{F}=\left \langle y^3, y^2, xyz\right \rangle$

Possible Answers:

$\left \langle -xz, yz, 3y^2\right \rangle$

$\left \langle xz, -yz, -3y^2\right \rangle$

$\left \langle xz, yz, 3y^2\right \rangle$

$\left \langle xyz, -yz, -3y^2\right \rangle$

$\vec{0}$

Correct answer:

$\left \langle xz, -yz, -3y^2\right \rangle$

Explanation:

The curl of the vector field is given by

$\begin{vmatrix} \hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ y^3&y^2 &xyz \end{vmatrix}$

$\frac{\partial }{\partial y}xyz\hat{i}+\frac{\partial }{\partial x}y^2\hat{k}+\frac{\partial }{\partial z}y^3\hat{j}-(\frac{\partial }{\partial y}y^3\hat{k}+\frac{\partial }{\partial z}y^2\hat{i}+\frac{\partial }{\partial x}xyz\hat{j})= xz\hat{i}-3y^2\hat{k}-yz\hat{j}=\left \langle xz, -yz, -3y^2\right \rangle$

Report an Error

Example Question #1744 : Calculus 3

Determine whether the vector field is conservative or not, and why:

$\vec{F}=\left \langle x^2, \ln(y^2), z^3 \right \rangle$

Possible Answers:

The vector field is not conservative because the curl equals zero.

The vector field is conservative because the curl equals zero.

The vector field is not conservative because the curl does not equal zero.

The vector field is conservative because the curl does not equal zero.

Correct answer:

The vector field is conservative because the curl equals zero.

Explanation:

A vector field is conservative if its curl produces the zero vector.

The curl of the vector field is given by

$\begin{vmatrix} \hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ x^2&\ln(y^2) &z^3 \end{vmatrix}$

$\frac{\partial }{\partial y}z^3\hat{i}+\frac{\partial }{\partial x}\ln(y^2)\hat{k}+\frac{\partial }{\partial z}x^2\hat{j}-(\frac{\partial }{\partial y}x^2\hat{k}+\frac{\partial }{\partial z}\ln(y^2)\hat{i}+\frac{\partial }{\partial x}z^3\hat{j}) = 0\hat{i}+0\hat{j}+0\hat{k}=\vec{0}$

The vector field is conservative.

Report an Error

View Calculus 3 Tutors

Thavisha
Certified Tutor

Minnesota State University-Mankato, Bachelor of Science, Mathematics.

View Calculus 3 Tutors

Jason
Certified Tutor

Brigham Young University, Bachelor, Neuroscience.

View Calculus 3 Tutors

Jeremy
Certified Tutor

University of Illinois at Chicago, Bachelor of Science, Physics.

All Calculus 3 Resources

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

Popular Subjects

ISEE Tutors in San Diego, English Tutors in Seattle, English Tutors in San Diego, Spanish Tutors in Dallas Fort Worth, Statistics Tutors in Denver, SSAT Tutors in Washington DC, GRE Tutors in New York City, GRE Tutors in Los Angeles, Physics Tutors in San Francisco-Bay Area, Statistics Tutors in Los Angeles

Popular Courses & Classes

Spanish Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Phoenix, ISEE Courses & Classes in Chicago, ISEE Courses & Classes in Los Angeles, SSAT Courses & Classes in Washington DC, SAT Courses & Classes in Washington DC, MCAT Courses & Classes in Philadelphia, GMAT Courses & Classes in San Diego, SAT Courses & Classes in Denver, LSAT Courses & Classes in Denver

Popular Test Prep

ACT Test Prep in Dallas Fort Worth, MCAT Test Prep in New York City, ISEE Test Prep in Los Angeles, ISEE Test Prep in Philadelphia, SAT Test Prep in Dallas Fort Worth, ISEE Test Prep in San Diego, GRE Test Prep in Phoenix, MCAT Test Prep in Los Angeles, ISEE Test Prep in Phoenix, ISEE Test Prep in Boston

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 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #1741 : Calculus 3

Example Question #1742 : Calculus 3

Example Question #20 : Curl

Example Question #21 : Curl

Example Question #91 : Line Integrals

Example Question #21 : Curl

Example Question #24 : Curl

Example Question #22 : Curl

Example Question #1743 : Calculus 3

Example Question #1744 : Calculus 3

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: