Thread: conservative and independent of path - line integral

1. conservative 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

2. 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

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

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 $\displaystyle g(y,z) = C)$

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

3. 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.
$\displaystyle f = x^2 y^3 z^4$.

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

Thank you very much.

5. 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.

6. 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.

7. 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.