The vector field F(x, y, z) = 2xi+zj+yk is conservative. Find a corresponding potential function. That is, find a function f(x, y, z) such that F(x, y, z) = (upside down triangle)f(x, y, z).

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- September 9th 2008, 05:30 PMwik_chick88corresponding potential function
The vector field F(x, y, z) = 2xi+zj+yk is conservative. Find a corresponding potential function. That is, find a function f(x, y, z) such that F(x, y, z) = (upside down triangle)f(x, y, z).

- September 9th 2008, 06:47 PMchoovuck
as I understand, we need to solve the system

Integrating first equation we obtain . Plugging to the second and integrating, we get , and then plugging that to the third equation and integrating, we get . Hence for any constant c - September 9th 2008, 06:55 PMJhevon
have you done differential equations before? this is done in much the same way as the method of exact equations. essentially, we are looking for a function so that .

so we have,

.............(1)

...............(2)

...............(3)

integrating (1) with respect to , we get

...........we think of our constant as a function of and , for (hopefully) obvious reasons

now, differentiate this with respect to

comparing this to (2), we see that

so that , by integrating with respect to , leaving a constant in terms of a funciton of .

so now, go back to , we see that,

differentiate this with respect to and compare to (3), we see that:

, a constant, so finally,

it is trivial to verify that