# [SOLVED] General potential of a field

• Mar 13th 2010, 10:19 AM
arbolis
[SOLVED] General potential of a field
Let $\displaystyle \bold K(x,y,z)=(4xyz,2x^2z+2yz^2,2x^2y+2y^2z+4z^3)$ defined in $\displaystyle \mathbb{R}^3$.
Show that the field is irrotational and calculate the general potential of this field.
I've calculated its curl (0), so the field is irrotational.
Now I must find, I believe, a vector field whose curl is $\displaystyle \bold K$. I called it $\displaystyle \bold A=(a_1,a_2,a_3)$ but I've too much unknowns. I don't know how to solve the last part. Can someone help me?

In details I wrote $\displaystyle \frac{\partial a_3}{\partial y}-\frac{\partial a_2}{\partial z}=4xyz$ and 2 other relations of the same kind. At last I ended up with at least 9 unknowns and only 3 equations.
• Mar 13th 2010, 10:34 AM
Random Variable
Since K is irrotational, it's the gradient of a function.
• Mar 13th 2010, 10:34 AM
Black
I believe you need to find $\displaystyle \varphi$ such that

$\displaystyle \mathbf{K} = -\nabla \varphi ,$ so you have

$\displaystyle \varphi_x = -4xyz$

$\displaystyle \varphi_y = -2x^2z-2yz^2$

$\displaystyle \varphi_z = -2x^2y-2y^2z-4z^3.$
• Mar 13th 2010, 10:48 AM
arbolis
Ah thanks to both. I get it. I forgot the theorem or consequence of a theorem that if a field is irrotational, then it's the gradient of some function.
• Mar 21st 2010, 02:48 PM
arbolis
Clueless
I thought I had it, but I realize I made a mistake.
Now I got it, but I think I got it by chance. Would someone explain me how to reach $\displaystyle \varphi$ from $\displaystyle \varphi _x$, $\displaystyle \varphi _y$ and $\displaystyle \varphi _z$?
I realized that $\displaystyle \int \varphi _z dz =\varphi$. Which gives $\displaystyle \varphi =-2x^2yz -y^2z^2-z^4$.

I was thinking about integrating $\displaystyle \varphi _x$ with respect to $\displaystyle dx$ and sum up a function $\displaystyle g(y,z)$ then integrating $\displaystyle \varphi _y$ with respect to y and sum up a function $\displaystyle g_1(x,z)$ and integrating $\displaystyle \varphi _z$ with respect to z and sum up a function $\displaystyle g_2 (x,y)$. Lastly, summing all these functions and try to determine $\displaystyle g$, $\displaystyle g_1$ and $\displaystyle g_2$.
I just don't understand how I got the right result with what I've done...
• Mar 21st 2010, 03:20 PM
Random Variable
That's what I would do.

$\displaystyle \varphi = -2x^{2}yz + g_{1}(y,z) = -2x^{2}yz-y^{2}z^{2} + g_{2}(x,z) = 2x^{2}yz - y^{2}z^{2}-z^{4} + g_{3}(x,y)$

so $\displaystyle g_{1}(y,z) = -y^{2}z^{2}-z^{4}, g_{2}(x,z) = -z^{4}, \text{and} \ g_{3}(x,y) =0$
• Mar 21st 2010, 03:22 PM
arbolis
Quote:

Originally Posted by Random Variable
That's what I would do.

$\displaystyle \varphi = -2x^{2}yz + g_{1}(y,z) = -2x^{2}yz-y^{2}z^{2} + g_{2}(x,z) = 2x^{2}yz - y^{2}z^{2}-z^{4} + g_{3}(x,y)$

so $\displaystyle g_{1}(y,z) = -y^{2}z^{2}-z^{4}, g_{2}(x,z) = -z^{4}, \text{and} \ g_{3}(x,y) =0$

Makes perfectly sense. (Clapping)