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## Homework Statement

Given a vector field defined by:

## \vec F = 2xc\hat x + cz\hat y + cx\hat z ##

Is conservative

Find a potential function for F. Verify this by taking the gradient to recover F.

## Homework Equations

**## V = - \int_a^b \vec F \cdot \vec dr ##**

## F = -\nabla V ##

## F = -\nabla V ##

## The Attempt at a Solution

The field is conservative, and this was found easily by computing the curl and finding it to be zero.

To find the potential function, I tried to simply apply the second formula in section two, as such:

## \vec dr = dx\hat x + dy\hat y + dz\hat z ##

And to choose a path from (0,0,0) to (x,y,z).

So then:

## \vec F \cdot \vec dr = 2xcdx + czdy + cydz ##

And the resulting integral is:

## - ( \int_0^x 2xcdx + \int_0^y czdy + \int_0^z cydz ) ##

Which results in:

## V = -cx^{2} - cyz - cyz = -cx^{2} - 2cyz ##

But when I try to get back to F using the negative gradient of V, I get the correct answer for the x component, but the factor of two ends up giving me the wrong answer for the y and z components. I'm not sure exactly what went wrong or how to fix it at this point.