When you have a conservative vector field, it is sometimes possible to calculate a potential function, i.e. to undo 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 multivariable (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.
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 \( \nabla f = \vec{F} \).
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 

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 

video by PatrickJMT 

close solution

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

Determine if the vector field \( \vec{F} = \langle 3x+2y, 2x3y \rangle \) is conservative. If it is, find the potential function.
Problem Statement 

Determine if the vector field \( \vec{F} = \langle 3x+2y, 2x3y \rangle \) is conservative. If it is, find the potential function.
Solution 

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 

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 

video by PatrickJMT 

close solution

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

Determine if the vector field \( \vec{F} = \langle z^2+2xy, x^2+2, 2xz1 \rangle \) is conservative. If it is, find the potential function.
Problem Statement 

Determine if the vector field \( \vec{F} = \langle z^2+2xy, x^2+2, 2xz1 \rangle \) is conservative. If it is, find the potential function.
Solution 

video by MIP4U 

close solution

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

You CAN Ace Calculus
external links you may find helpful 

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  doubleangle formulas  

\( \sin(2t) = 2\sin(t)\cos(t)\) 
\(\displaystyle{ \cos(2t) = \cos^2(t)  \sin^2(t) }\) 
Set 4  halfangle 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
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\) 
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\)  
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) 
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = \frac{1}{1+t^2} }\)  
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
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.
Single Variable Calculus 

MultiVariable Calculus 

Differential Equations 

Precalculus 

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.
 
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. 


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