\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus - Potential Functions

Coordinate Systems

Vectors

Using Vectors

Applications

Vector Functions

Partial Derivatives

Partial Integrals

Double Integrals - 2Int

Triple Integrals - 3Int

Practice

Vector Fields

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Tools

Calculus Tools

Additional Tools

Articles

Coordinate Systems

Vectors

Using Vectors

Applications

Vector Functions

Partial Derivatives

Partial Integrals

Double Integrals - 2Int

Triple Integrals - 3Int

Practice

Vector Fields

SV Calculus

MV Calculus

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Tools

Calculus Tools

Additional Tools

Articles

When you have a conservative vector field, it is sometimes possible to calculate a potential function, i.e. to un-do the gradient.

If you have a conservative vector field, you will probably be asked to determine the potential function. This is the function from which conservative vector field ( the gradient ) can be calculated. So you just need to set up two or three multi-variable (partial) integrals (depending if you are working in \( \mathbb{R}^2\) or \( \mathbb{R}^3 \)), evaluate them and combine them to get one potential function. The equations look like this.

given conservative vector field ( a gradient )

\( \vec{G}(x,y,z) = G_i\hat{i} + G_j\hat{j} + G_k\hat{k} \)

potential function to calculate using \( \nabla g = \vec{G}\)

\( g(x,y,z) \)

\( \partial g / \partial x = G_i ~~~ \to ~~~ \int{G_i~dx} = g_1 + c_1(y,z) \)

\( \partial g / \partial y = G_j ~~~ \to ~~~ \int{G_j~dy} = g_2 + c_2(x,z) \)

\( \partial g / \partial z = G_k ~~~ \to ~~~ \int{G_k~dz} = g_3 + c_3(x,y) \)

combine \( g_1 + c_1(y,z) \), \( g_2 + c_2(x,z) \) and \( g_3 + c_3(x,y) \) to get \(g(x,y,z)\)

practice problem 841 shows, in detail, how this works

Here is a great video, with examples, showing how this works.

Dr Chris Tisdell - Potential Function Example (Gradient) [5mins-19secs]

video by Dr Chris Tisdell

Practice

Unless otherwise instructed, determine if the given vector field is conservative. If it is, find the potential function.

Basic

Determine if the vector field \( \vec{F}(x,y,z) = 2xy\vhat{i} + (x^2+2yz)\vhat{j} + y^2 \vhat{k} \) is conservative. If it is, find the potential function.

Problem Statement

Determine if the vector field \( \vec{F}(x,y,z) = 2xy\vhat{i} + (x^2+2yz)\vhat{j} + y^2 \vhat{k} \) is conservative. If it is, find the potential function.

Solution

The video does not show how to determine the potential function, so we include the details here.

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

\( \int{2xy~dx} = x^2y + c_1(y,z) \)

\( \int{x^2+2yz~dy} = x^2y + y^2z + c_2(x,z) \)

\( \int{y^2~dz} = y^2z + c_3(x,y) \)

Now we need to combine the results of the three integrals.
From the first integration, we have \(x^2y\).
From the second integration, we have \( y^2z \). We do not use \(x^2y\) since we got it from the first integration.
From the third integration, we have nothing new. The term \( y^2z \) we already have from the second integration.
So our potential function is \( f(x,y,z) = x^2y + y^2z \). Technically, we should have a constant in the equation but we usually let that be equal to zero. So we should really call this A potential function, one of many, in fact infinite, from which we could determine the conservative vector field. With a quick calculation in your head, you can verify that \( \nabla f = \vec{F} \).

841 video

video by PatrickJMT

close solution

Log in to rate this practice problem and to see it's current rating.

Determine if the vector field \( \vec{F} = x^2\hat{i} + y\hat{j} \) is conservative. If it is, find the potential function.

Problem Statement

Determine if the vector field \( \vec{F} = x^2\hat{i} + y\hat{j} \) is conservative. If it is, find the potential function.

Solution

849 video

video by Dr Chris Tisdell

close solution

Log in to rate this practice problem and to see it's current rating.

Determine if the vector field \( \vec{F} = \langle 2x+yz, xz, xy \rangle \) is conservative. If it is, find the potential function.

Problem Statement

Determine if the vector field \( \vec{F} = \langle 2x+yz, xz, xy \rangle \) is conservative. If it is, find the potential function.

Solution

850 video

video by PatrickJMT

close solution

Log in to rate this practice problem and to see it's current rating.

Determine if the vector field \( \vec{F} = \langle 3x+2y, 2x-3y \rangle \) is conservative. If it is, find the potential function.

Problem Statement

Determine if the vector field \( \vec{F} = \langle 3x+2y, 2x-3y \rangle \) is conservative. If it is, find the potential function.

Solution

852 video

video by MIP4U

close solution

Log in to rate this practice problem and to see it's current rating.

Determine if the vector field \( \nabla f = (2x+yz)\hat{i} + xz\hat{j} + xy\hat{k} \) is conservative. If it is, find the potential function.

Problem Statement

Determine if the vector field \( \nabla f = (2x+yz)\hat{i} + xz\hat{j} + xy\hat{k} \) is conservative. If it is, find the potential function.

Solution

811 video

video by PatrickJMT

close solution

Log in to rate this practice problem and to see it's current rating.

Intermediate

Determine if the vector field \( \vec{F} = (6xy+4z^2)\hat{i} + (3x^2+3y^2)\hat{j} + 8xz\hat{k} \) is conservative. If it is, find the potential function.

Problem Statement

Determine if the vector field \( \vec{F} = (6xy+4z^2)\hat{i} + (3x^2+3y^2)\hat{j} + 8xz\hat{k} \) is conservative. If it is, find the potential function.

Solution

851 video

video by PatrickJMT

close solution

Log in to rate this practice problem and to see it's current rating.

Determine if the vector field \( \vec{F} = \langle z^2+2xy, x^2+2, 2xz-1 \rangle \) is conservative. If it is, find the potential function.

Problem Statement

Determine if the vector field \( \vec{F} = \langle z^2+2xy, x^2+2, 2xz-1 \rangle \) is conservative. If it is, find the potential function.

Solution

853 video

video by MIP4U

close solution

Log in to rate this practice problem and to see it's current rating.

You CAN Ace Calculus

Topics You Need To Understand For This Page

Related Topics and Links

external links you may find helpful

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

To bookmark this page and practice problems, log in to your account or set up a free account.

Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

learning and study techniques

Get great tutoring at an affordable price with Chegg. Subscribe today and get your 1st 30 minutes Free!

The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.

How to Ace the Rest of Calculus: The Streetwise Guide, Including MultiVariable Calculus

Shop eBags.com, the leading online retailer of luggage, handbags, backpacks, accessories, and more!

Try Amazon Music Unlimited Free Trial

Math Word Problems Demystified

Shop eBags.com, the leading online retailer of luggage, handbags, backpacks, accessories, and more!

Shop Amazon - Sell Us Your Books - Get up to 80% Back

Practice Instructions

Unless otherwise instructed, determine if the given vector field is conservative. If it is, find the potential function.

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.

17calculus

Copyright © 2010-2020 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics
17Calculus
We use cookies to ensure that we give you the best experience on our website. By using this site, you agree to our Website Privacy Policy.