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Conservative Vector Fields and Potential Functions |
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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 |
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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 DimensionsIn 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. |
Okay, let's watch a quick video that explains this idea.
PatrickJMT - Conservative Vector Fields - The Definition and a Few Remarks | |
Potential Function |
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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 A01 below shows, in detail, how this works |
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Practice Problems |
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Instructions - - Unless otherwise instructed, determine if the given vector field is conservative. If it is, find the potential function.
Level A - Basic |
Practice A01 | |
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\( \vec{F}(x,y,z) = 2xy\vhat{i} + (x^2+2yz)\vhat{j} + y^2 \vhat{k} \) | |
solution |
Practice A02 | |
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\( \vec{F} = x^2\hat{i} + y\hat{j} \) | |
solution |
Practice A03 | |
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\( \vec{F} = \langle 2x+yz, xz, xy \rangle \) | |
solution |
Practice A04 | |
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\( \vec{F} = \langle 3x+2y, 2x-3y \rangle \) | |
solution |
Level B - Intermediate |
Practice B01 | |
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\(\vec{F}=(6xy+4z^2)\hat{i} + (3x^2+3y^2)\hat{j} + 8xz\hat{k} \) | |
solution |
Practice B02 | |
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\( \vec{F} = \langle z^2+2xy, x^2+2, 2xz-1 \rangle \) | |
solution |