Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Divergence

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 #31 : Divergence

Compute the divergence of the following vector function:

$\mathbf{F}=\cos(xy)\mathbf{i}+4y^3z^2\mathbf{j}+5xe^{2z}\mathbf{k}$

Possible Answers:

$-y\sin(xy)+12y^2z^2+10xe^{2z}$

$-\sin(xy)+12y^2z^2+5xe^{2z}$

$y\sin(x)+12y^2z^3+10xe^{z}$

$-x\cos(xy)+16y^2z^2+5e^{2z}$

Correct answer:

$-y\sin(xy)+12y^2z^2+10xe^{2z}$

Explanation:

For a vector function $\mathbf{F}=U\mathbf{i}+V\mathbf{j}+W\mathbf{k}$ ,

the divergence is defined by:

$\nabla \cdot \mathbf{F}=\frac{\partial U}{\partial x}+\frac{\partial V}{\partial y} +\frac{\partial W}{\partial z}$

For our function:

$\frac{\partial U}{\partial x}=\frac{\partial }{\partial x}(\cos(xy))=-y\sin(xy)$

$\frac{\partial V}{\partial y}=\frac{\partial }{\partial y}(4y^3z^2)=12y^2z^2$

$\frac{\partial W}{\partial z}=\frac{\partial }{\partial z}(5xe^{2z})=10xe^{2z}$

Thus, the divergence of our function is:

$\nabla \cdot \mathbf{F}=-y\sin(xy)+12y^2z^2+10xe^{2z}$

Report an Error

Example Question #32 : Divergence

Find $div \vec{F}$ , where F is given by the following:

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

Possible Answers:

$\frac{1}{x}+3z^2\cos(z)-z^3\sin(z)$

$\frac{1}{x}+3z^2\cos(z)+z^3\sin(z)$

$\left \langle 0, \frac{1}{x}, 3z^2\cos(z)-z^3\sin(z)\right \rangle$

Correct answer:

$\frac{1}{x}+3z^2\cos(z)-z^3\sin(z)$

Explanation:

The divergence of a vector function is given by

$div \vec{F}= \bigtriangledown \cdot \vec{F}$

where $\bigtriangledown = \left \langle f_x, f_y, f_z\right \rangle$

So, we must find the partial derivative of each respective component.

The partial derivatives are

f_x=0

$f_y=\frac{1}{x}$

$f_z=3z^2\cos(z)-z^3\sin(z)$

The partial derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$

Report an Error

Example Question #33 : Divergence

Find $div\vec{F}$ , where F is given by the following curve:

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

Possible Answers:

$2y+3y^2-xz^2\sin(xy)$

$3y^2+2z\cos(xy)$

$\left \langle 0, 3y^2, 2z\cos(xy)\right \rangle$

Correct answer:

$3y^2+2z\cos(xy)$

Explanation:

The divergence of a curve is given by

$div\vec{F}=\nabla \cdot \vec{F}$

where $\nabla = \left \langle f_x, f_y, f_z\right \rangle$

So, we must find the partial derivatives of the x, y, and z components, respectively:

f_x=0

f_y=3y^2

$f_z=2z\cos(xy)$

The partial derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Report an Error

Example Question #34 : Divergence

Find $div \vec{F}$ of the given function:

$\vec{F}=\left \langle 4xy, y\cos(y), xy\right \rangle$

Possible Answers:

$4y+\cos(y)+y\cos(y)$

$\cos(y)+y(4+\sin(y))$

$\cos(y)+y(4-\sin(y))$

$\cos(y)+4-\sin(y)$

Correct answer:

$\cos(y)+y(4-\sin(y))$

Explanation:

The divergence of a vector function is given by

$div \vec{F}=\bigtriangledown \cdot \vec{F}$

where $\bigtriangledown = \left \langle f_x, f_y, f_z\right \rangle$

So, we take the respective partial derivatives of the x, y, and z-components of the vector function, and add them together (from the dot product):

f_x=4y

$f_y=\cos(y)-y\sin(y)$

f_z=0

The partial derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$

Report an Error

Example Question #35 : Divergence

Find $div \vec{F}$ of the vector function below:

$\vec{F}=\cos(x)\hat{i}+(xyz)\hat{j}+z^2\sin(y)\hat{k}$

Possible Answers:

$\left \langle -\sin(x),xz,2z\sin(y) \right \rangle$

$-\sin(x)+xz+2z\sin(y)$

$\sin(x)+xz+2z\sin(y)$

$-\sin(x)+xz-2z\sin(y)$

Correct answer:

$-\sin(x)+xz+2z\sin(y)$

Explanation:

The divergence of a vector function is given by

$div \vec{F}=\bigtriangledown \cdot \vec{F}$

where $\bigtriangledown = \left \langle f_x, f_y, f_z\right \rangle$

So, in taking the dot product of the gradient and the function, we get the sum of the respective partial derivatives. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

$f_x=-\sin(x)$

f_y=xz

$f_z=2z\sin(y)$

Report an Error

Example Question #36 : Divergence

Find $div \vec{F}$ of the vector function below:

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

Possible Answers:

$e^{\cos(y)+\sin(z)}+2y+1$

$e^{\cos(y)+\sin(z)}$

$e^{\cos(y)+\sin(z)}+2$

$e^{\cos(y)+\sin(z)}+3$

Correct answer:

$e^{\cos(y)+\sin(z)}+3$

Explanation:

The divergence of a vector function is given by

$div \vec{F}=\bigtriangledown \cdot \vec{F}$

where $\bigtriangledown = \left \langle f_x, f_y, f_z\right \rangle$

The partial derivatives are

$f_x=e^{\cos(y)+\sin(z)}$

f_y=2

f_z=1

Report an Error

Example Question #37 : Divergence

Find $div \vec{F}$ where F is given by

$\vec{F}=\left \langle x^2\cos(xy), yz^3, xy\right \rangle$

Possible Answers:

$2x\cos(xy)-x^2y\sin(xy)+z^3$

$\left \langle 2x\cos(xy), -x^2y\sin(xy), z^3 \right \rangle$

$2x\cos(xy)-x^2y\sin(xy)+z^3+xy$

$2x\cos(xy)+x^2y\sin(xy)+z^3$

Correct answer:

$2x\cos(xy)-x^2y\sin(xy)+z^3$

Explanation:

The divergence of a vector field is given by

$div \vec{F}=\bigtriangledown \cdot \vec{F}$

where $\bigtriangledown = \left \langle f_x, f_y, f_z\right \rangle$

In taking the dot product of the gradient and the vector field, we get the sum of the respective partial derivatives of F.

The partial derivatives are

$f_x=2x\cos(xy)-x^2y\sin(xy)$

f_y=z^3

f_z=0

Report an Error

Example Question #31 : Divergence

Find $div \vec{F}$ where F is given by

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

Possible Answers:

$-\frac{y}{x}+2x^2yz-\frac{1}{z}$

$\frac{1}{x}+2x^2yz+\frac{1}{z}$

$\frac{y}{x}+2xy^2z+\frac{1}{z}$

$\frac{y}{x}+2x^2yz+\frac{1}{z}$

Correct answer:

$\frac{y}{x}+2x^2yz+\frac{1}{z}$

Explanation:

The divergence of a vector field is given by

$div \vec{F}=\bigtriangledown \cdot \vec{F}$

where $\bigtriangledown = \left \langle f_x, f_y, f_z\right \rangle$

In taking the dot product of the gradient and the vector field, we get the sum of the respective partial derivatives of F.

The partial derivatives are

$f_x=\frac{y}{x}$ , f_y=2x^2yz , $f_z=\frac{1}{z}$

Report an Error

Example Question #31 : Divergence

Find $div \vec{F}$ where F is given by

$\vec{F}=\left \langle \sec(5x), \ln(z), xyz\right \rangle$

Possible Answers:

$-5\sec(5x)\tan(5x)+\frac{1}{z}+xy$

$5\sec(5x)\tan(5x)+\frac{1}{z}+xyz$

$\sec(5x)\tan(5x)+\frac{1}{z}+xy$

$5\sec(5x)\tan(5x)+\frac{1}{z}+xy$

Correct answer:

$5\sec(5x)\tan(5x)+\frac{1}{z}+xy$

Explanation:

The divergence of a vector field is given by

$div \vec{F}=\bigtriangledown \cdot \vec{F}$

where $\bigtriangledown = \left \langle f_x, f_y, f_z\right \rangle$

In taking the dot product of the gradient and the vector field, we get the sum of the respective partial derivatives of F.

The partial derivatives are

$f_x=5\sec(5x)\tan(5x)$

$f_y=\frac{1}{z}$

f_z=xy

Report an Error

Example Question #40 : Divergence

Find the divergence of the vector $\left \langle 2x^2y^3,y^5,\cos(z)\right \rangle$

Possible Answers:

$4xy^3+5y^4-\sin(z)$

$xy^3+5y^4-\sin(z)$

$4xy^3+7y^4+\sin(z)$

$4xy^2+7y^4-\sin(z)$

Correct answer:

$4xy^3+5y^4-\sin(z)$

Explanation:

To find the divergence of a vector $a=\left \langle A,B,C\right \rangle$ , we use the definition

$divergence=\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}$

Using the vector from the problem statement, we get

$divergence=\frac{\partial }{\partial x}(2x^2y^3)+\frac{\partial }{\partial y}(y^5)+\frac{\partial }{\partial z}(\cos(z))=4xy^3+5y^4-\sin(z)$

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

Muhammad
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Mechanical Engineering.

View Calculus 3 Tutors

Dawn
Certified Tutor

University of Oregon, Bachelor in Arts, Biochemistry and Molecular Biology. Johns Hopkins University, Doctor of Philosophy, E...

All Calculus 3 Resources

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

Popular Subjects

MCAT Tutors in Denver, Calculus Tutors in Houston, GMAT Tutors in Dallas Fort Worth, Biology Tutors in San Diego, Biology Tutors in Seattle, Computer Science Tutors in Los Angeles, LSAT Tutors in Miami, French Tutors in San Francisco-Bay Area, SAT Tutors in Philadelphia, Biology Tutors in Houston

Popular Courses & Classes

LSAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Miami, Spanish Courses & Classes in San Diego, Spanish Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Denver, MCAT Courses & Classes in Philadelphia, SSAT Courses & Classes in New York City, GMAT Courses & Classes in Philadelphia, ISEE Courses & Classes in Phoenix, GRE Courses & Classes in Houston

Popular Test Prep

MCAT Test Prep in Dallas Fort Worth, MCAT Test Prep in San Francisco-Bay Area, MCAT Test Prep in Chicago, GMAT Test Prep in Phoenix, GMAT Test Prep in Chicago, ISEE Test Prep in Chicago, ISEE Test Prep in Atlanta, ACT Test Prep in Miami, SSAT Test Prep in Dallas Fort Worth, GRE Test Prep in Seattle

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 : Divergence

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #31 : Divergence

Example Question #32 : Divergence

Example Question #33 : Divergence

Example Question #34 : Divergence

Example Question #35 : Divergence

Example Question #36 : Divergence

Example Question #37 : Divergence

Example Question #31 : Divergence

Example Question #31 : Divergence

Example Question #40 : Divergence

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: