## 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.

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

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

### Practice

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 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 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 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 video

video by MIP4U

$$\nabla f = (2x+yz)\hat{i} + xz\hat{j} + xy\hat{k}$$

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

### 811 video

video by PatrickJMT

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 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

### conservative vector fields and potential functions 17calculus youtube playlist

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