\( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \)
Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Radius of Convergence
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Arc Length
Surface Area
Polar Coordinates
Slope & Tangent Lines
Arc Length
Surface Area
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Ready For Calculus 2?
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Practice Exams
17calculus on YouTube
More Math Help
Tools and Resources
Academic Integrity
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Instructor or Coach?
Math Books
How To Read Math Books

You CAN Ace Calculus

17calculus > vector fields > conservative vector fields

ATTENTION INSTRUCTORS: The new 2018 version of 17calculus will include changes to the practice problem numbering system. If you would like advance information to help you prepare for spring semester, send us an email at 2018info at 17calculus.com.

free ideas to save on books - bags - supplies

math and science learning techniques

Join Amazon Student - FREE Two-Day Shipping for College Students

How to Develop a Brilliant Memory Week by Week: 50 Proven Ways to Enhance Your Memory Skills

Conservative Vector Fields and Potential Functions

A conservative vector field may also be called a gradient field.

A conservative vector field is a vector field that is a gradient of some function, in this context called a potential function. The idea is that you are given a gradient and you have to 'un-gradient' it to get the original function. The only kind of vector fields that you can 'un-gradient' are conservative vector fields.

Test For Conservative Vector Field

Conservative vector fields are also called irrotational since the curl is zero. So, you now have a test to see if a vector field is conservative: calculate the curl and see if it's zero. If so, then it is conservative, otherwise it is not conservative. Okay, so you may be saying to yourself, that works for three dimensions, but what about two dimensions? Well, it works for two dimensions too but your instructor may give you a 'special' equation. The next panel discusses this 'special' equation.

Conservative Vector Fields in Two Dimensions

In the above discussion, we said that the test for a conservative vector field is to calculate the curl to see if it is zero. However, the curl uses the idea of the cross product, which works only in three dimensions. Here is how we apply the same idea to a two dimensional situation. This frees you from having to memorize those 'special' equations.

Let's say you are given the two dimensional vector field \( \vec{F}(x,y) = M(x,y)\hat{i} + N(x,y)\hat{j} \) and you need to determine if it is conservative.
First, we expand it to three dimensions so that we can use find the curl. So we rewrite it as \( \vec{G}(x,y) = M(x,y)\hat{i} + N(x,y)\hat{j} + 0\hat{k} \). We renamed this new three dimensional vector as \(\vec{G}\) just to be clear that we are now working in three dimensions. Okay, let's set up the curl and evaluate it.

\(\displaystyle{ \begin{array}{rcl} \vec{ \nabla } \times \vec{G} & = & \begin{vmatrix} \vhat{i} & \vhat{j} & \vhat{k} \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ M(x,y) & N(x,y) & 0 \end{vmatrix} = \begin{vmatrix} \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ N(x,y) & 0 \end{vmatrix} \hat{i} - \begin{vmatrix} \frac{\partial }{\partial x} & \frac{\partial }{\partial z} \\ M(x,y) & 0 \end{vmatrix} \hat{j} + \begin{vmatrix} \frac{\partial }{\partial x} & \frac{\partial }{\partial y} \\ M(x,y) & N(x,y) \end{vmatrix} \hat{k} \\ \\ & = & \left[ \frac{\partial 0 }{\partial y} - \frac{\partial N(x,y)}{\partial z} \right] \hat{i} - \left[ \frac{\partial 0 }{\partial x} - \frac{\partial M(x,y)}{\partial z} \right] \hat{j} + \left[ \frac{\partial N(x,y)}{\partial x} - \frac{\partial M(x,y)}{\partial y} \right] \hat{k} \end{array} }\)

Okay, let's look at each component separately. Of course, you know that the derivative of zero is zero. That's easy. So for the \(\hat{i}\) component we have the term \( \partial N(x,y) / \partial z \). Notice that \(N(x,y)\) does not contain any \(z\)'s. So this derivative is zero. The same reasoning holds for \( \partial M(x,y) / \partial z \). This means that we have zero for both the \(\hat{i}\) and \(\hat{j}\) components.

For a vector field to be conservative, we need the curl to be zero. Since we have zero for both the \(\hat{i}\) and \(\hat{j}\) components, we just need to look at the \(\hat{k}\) component. For it to be zero, we need

\(\displaystyle{ \frac{\partial N(x,y)}{\partial x} = \frac{\partial M(x,y)}{\partial y} }\)

And that's your 'special' equation. Cool, eh? We don't think that special equations should be memorized. We recommend that you understand the technique that applies to a wider range of problems. However, what your instructor expects comes first. So check with them before you take our advice.

Okay, let's watch a quick video that explains this idea.

PatrickJMT - Conservative Vector Fields - The Definition and a Few Remarks

Potential Function

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 A01 below shows, in detail, how this works

Search 17Calculus

Practice Problems

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

Level A - Basic

Practice A01

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


Practice A02

\( \vec{F} = x^2\hat{i} + y\hat{j} \)


Practice A03

\( \vec{F} = \langle 2x+yz, xz, xy \rangle \)


Practice A04

\( \vec{F} = \langle 3x+2y, 2x-3y \rangle \)


Level B - Intermediate

Practice B01

\(\vec{F}=(6xy+4z^2)\hat{i} + (3x^2+3y^2)\hat{j} + 8xz\hat{k} \)


Practice B02

\( \vec{F} = \langle z^2+2xy, x^2+2, 2xz-1 \rangle \)


Real Time Web Analytics
menu top search practice problems
menu top search practice problems 17