conservative and independent of path - line integral

**kittycat** · November 10th 2007, 02:45 AM

question.

Find a potential function for the vector field

F = 2xy^3z^4 i + 3x^2 y^2 z^4 j + 4x^2 y^3 z^3 k

I don't know how to do this question. Please help me . Thank you very much.

**kalagota** · November 10th 2007, 03:32 AM

Originally Posted by kittycat

question.

Find a potential function for the vector field

F = 2xy^3z^4 i + 3x^2 y^2 z^4 j + 4x^2 y^3 z^3 k

I don't know how to do this question. Please help me . Thank you very much.

$F = 2xy^3z^4 i + 3x^2 y^2 z^4 j + 4x^2 y^3 z^3 k$

note that $F = 2xy^3z^4 i + 3x^2 y^2 z^4 j + 4x^2 y^3 z^3 k = \bigtriangledown \Phi(x,y,z)$

so:
$2xy^3z^4 = \frac{\partial \Phi}{\partial x}$

$3x^2 y^2 z^4 = \frac{\partial \Phi}{\partial y}$

$4x^2 y^3 z^3 = \frac{\partial \Phi}{\partial z}$

$\implies \Phi (x,y,z) = x^2y^3z^4 + g(y,z)$

from here,
$\frac{\partial \Phi}{\partial y} = 3x^2y^2z^4 + \frac{\partial g}{\partial y}$ and $\frac{\partial \Phi}{\partial y} = 4x^2y^3z^3 + \frac{\partial g}{\partial z}$

$\implies \frac{\partial g}{\partial y} = 0$ and $\frac{\partial g}{\partial z} = 0$ (why?)

$\implies g(y,z) = c_1$ and $g(y,z) = c_2$ (or $g(y,z) = C)$

$\implies \Phi (x,y,z) = x^2y^3z^4 + C$

**Plato** · November 10th 2007, 03:32 AM

Check to see if it is conservative.
$\begin{array}{l}<br /> J_z = 12x^2 y^2 z^3 = K_y \\ <br /> K_x = 8xy^3 z^3 = I_z \\ <br /> J_x = 6xy^2 z^4 = I_y \\ <br /> \end{array}$
It is.

Then find the primitive function.
$f = x^2 y^3 z^4$ .

**kittycat** · November 10th 2007, 07:11 AM

hi kalagota,
Why + g(y,z) ??? Could you please explain this point to me?

Thank you very much.

**Plato** · November 10th 2007, 09:45 AM

Originally Posted by kittycat

Why + g(y,z) ??? Could you please explain this point to me?

The g(y,z) would act as a constant function when differentiating with respect to x.

**ThePerfectHacker** · November 10th 2007, 01:52 PM

1) Kalagota use \nabla instead of the monster you are using.

2) You can see if curl is a zero vector to know if it has a scalar potentional.

**Plato** · November 10th 2007, 02:03 PM

Originally Posted by ThePerfectHacker

You can see if curl is a zero vector to know if it has a scalar potentional.

If you note, that is exactly what I did in my solution.

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