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.

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- Nov 10th 2007, 02:45 AMkittycatconservative and independent of path - line integral
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. - Nov 10th 2007, 03:32 AMkalagota
$\displaystyle F = 2xy^3z^4 i + 3x^2 y^2 z^4 j + 4x^2 y^3 z^3 k$

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

so:

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

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

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

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

from here,

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

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

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

$\displaystyle \implies \Phi (x,y,z) = x^2y^3z^4 + C$ - Nov 10th 2007, 03:32 AMPlato
Check to see if it is conservative.

$\displaystyle \begin{array}{l}

J_z = 12x^2 y^2 z^3 = K_y \\

K_x = 8xy^3 z^3 = I_z \\

J_x = 6xy^2 z^4 = I_y \\

\end{array}$

It is.

Then find the primitive function.

$\displaystyle f = x^2 y^3 z^4 $. - Nov 10th 2007, 07:11 AMkittycat
http://www.mathhelpforum.com/math-he...84bed1eb-1.gif

hi kalagota,

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

Thank you very much. - Nov 10th 2007, 09:45 AMPlato
- Nov 10th 2007, 01:52 PMThePerfectHacker
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. - Nov 10th 2007, 02:03 PMPlato