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17Calculus - Potential Functions

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

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

Math Word Problems Demystified

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 \( \nabla f = \vec{F} \).

PatrickJMT - 841 video solution

video by PatrickJMT

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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

Dr Chris Tisdell - 849 video solution

video by Dr Chris Tisdell

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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

PatrickJMT - 850 video solution

video by PatrickJMT

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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

MIP4U - 852 video solution

video by MIP4U

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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

PatrickJMT - 811 video solution

video by PatrickJMT

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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

PatrickJMT - 851 video solution

video by PatrickJMT

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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

MIP4U - 853 video solution

video by MIP4U

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

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