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### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

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17calculus > vector fields > conservative vector fields & potential functions

 Test For Conservative Vector Field Conservative Vector Fields in Two Dimensions Potential Function Practice
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. Let's watch a quick video that explains this idea in more detail.

### PatrickJMT - Conservative Vector Fields - The Definition and a Few Remarks [4mins-1sec]

video by PatrickJMT

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 section 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.

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

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.

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.

$$\partial g / \partial x = G_i ~~~ \to ~~~ \int{G_i~dx} = g_1 + c_1(y,z)$$ $$\partial g / \partial y = G_j ~~~ \to ~~~ \int{G_j~dy} = g_2 + c_2(x,z)$$ 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 $$\nabla g = \vec{G}$$ $$g(x,y,z)$$ 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

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

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

Basic Problems

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

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 solution video

video by PatrickJMT

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

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 solution video

video by Dr Chris Tisdell

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

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 solution video

video by PatrickJMT

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

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 solution video

video by MIP4U

Intermediate Problems

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

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 solution video

video by PatrickJMT

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

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

video by MIP4U