17Calculus - Potential 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 \( \nabla g = \vec{G}\) |
\( g(x,y,z) \) |
\( \partial g / \partial x = G_i ~~~ \to ~~~ \int{G_i~dx} = g_1 + c_1(y,z) \) | |
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\( \partial g / \partial y = G_j ~~~ \to ~~~ \int{G_j~dy} = g_2 + c_2(x,z) \) | |
\( \partial g / \partial z = G_k ~~~ \to ~~~ \int{G_k~dz} = g_3 + c_3(x,y) \) | |
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 |
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Practice
Instructions - - Unless otherwise instructed, determine if the given vector field is conservative. If it is, find the potential function.
Basic Problems |
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Problem Statement |
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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 |
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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 |
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Problem Statement |
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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.
Solution |
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849 video
video by Dr Chris Tisdell |
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Problem Statement |
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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.
Solution |
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850 video
video by PatrickJMT |
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Problem Statement |
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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.
Solution |
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852 video
video by MIP4U |
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Problem Statement |
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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.
Solution |
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811 video
video by PatrickJMT |
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Intermediate Problems |
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Problem Statement |
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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 |
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851 video
video by PatrickJMT |
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Problem Statement |
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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.
Solution |
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853 video
video by MIP4U |
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