# Conservative vector field

• May 7th 2013, 09:16 AM
Brennox
Conservative vector field
Hey, id love some help in doing this questions. Cheers
• May 7th 2013, 10:48 AM
topsquark
Re: Conservative vector field
Quote:

Originally Posted by Brennox
Hey, id love some help in doing this questions. Cheers

1. Integrate $\frac{\partial \phi}{\partial x} = 4y~cos(4xy)$ over x. This will be a function f(x, y, z) plus some function g(y, z).
For example, let $\frac{\partial \phi}{\partial x} = 3x + 4xy \implies \phi = \frac{3}{2}x^2 + 2x^2y + g(y, z)$
The function g is a constant as far as the integration is concerned, so we have g(y, z) as an arbitrary "constant."

Problems 2 and 3 use the same idea. See if you can get those for now.

-Dan
• May 7th 2013, 10:52 AM
Plato
Re: Conservative vector field
Quote:

Originally Posted by Brennox
Hey, id love some help in doing this questions. Cheers

Here is a sure fire way to reconstruct the primitive of a conservative field.

Let $I(x,y,z)=4y\cos(4xy),~J(x,y,z)=4x\cos(4xy),~\&~K(x ,y,z)=3z^2$ then

$\nu = \int_0^x {I(t,0,0)dt} + \int_0^y {J(x,t,0)dt} + \int_0^z {K(x,y,t)dt}$
• May 7th 2013, 11:04 PM
Brennox
Re: Conservative vector field
so i got sin(4xy) once integrated with respect to x. How do i get g(y,z)?

Quote:

Originally Posted by topsquark
1. Integrate $\frac{\partial \phi}{\partial x} = 4y~cos(4xy)$ over x. This will be a function f(x, y, z) plus some function g(y, z).
For example, let $\frac{\partial \phi}{\partial x} = 3x + 4xy \implies \phi = \frac{3}{2}x^2 + 2x^2y + g(y, z)$
The function g is a constant as far as the integration is concerned, so we have g(y, z) as an arbitrary "constant."

Problems 2 and 3 use the same idea. See if you can get those for now.

-Dan

• May 7th 2013, 11:34 PM
Brennox
Re: Conservative vector field
Quote:

Originally Posted by Plato
Here is a sure fire way to reconstruct the primitive of a conservative field.

Let $I(x,y,z)=4y\cos(4xy),~J(x,y,z)=4x\cos(4xy),~\&~K(x ,y,z)=3z^2$ then

$\nu = \int_0^x {I(t,0,0)dt} + \int_0^y {J(x,t,0)dt} + \int_0^z {K(x,y,t)dt}$

so i got sin(4xy) for 1st, sin(4xy) for 2nd, z^3 for 3rd and 2sin(4xy) + z^3 for a final answer. But i got it wrong, where am i wrong?
• May 8th 2013, 02:14 AM
Plato
Re: Conservative vector field
Quote:

Originally Posted by Brennox
so i got sin(4xy) for 1st, sin(4xy) for 2nd, z^3 for 3rd and 2sin(4xy) + z^3 for a final answer. But i got it wrong, where am i wrong?

The fist one is 0, so the final answer is $\sin(4xy)+z^3~.$