- #1

sylent33

- 39

- 5

- Homework Statement
- Find the pontential function of a vector field if possible

- Relevant Equations
- Jacobi Matrix

Hello! So I need to find the potential function of this Vector field

$$

\begin{matrix}

2xy -yz\\

x^2-xz\\

2z-xy

\end{matrix}

$$

Now first I tried to check if rotation is not ,since that is mandatory for the potentialfunction to exist.For that I used the jacobi matrix,and it was not symmetric hence rot cannot be 0.

Now to get the corresponding potentialfunction I first integrated the first row.It should look like this.

$$ x^2y-yzx+C(y,z) $$

Now this should be my corresponding x component (if we look at it like a vector) Now to find y I derrived this in respect to y and I got this

$$ x^2-zx + Cy(y,z) = x^2-zx $$

We can cancel a few things out and are left with

$$ Cy(y,z) = 1 $$

Now how do I get my y (or z) constant from this.I have no y or z standing anywhere and I think I made a mistake somwhere.Am I doing something wrong or should I simply procced like I would usually (which is to integrate both sides)Thank you!

$$

\begin{matrix}

2xy -yz\\

x^2-xz\\

2z-xy

\end{matrix}

$$

Now first I tried to check if rotation is not ,since that is mandatory for the potentialfunction to exist.For that I used the jacobi matrix,and it was not symmetric hence rot cannot be 0.

Now to get the corresponding potentialfunction I first integrated the first row.It should look like this.

$$ x^2y-yzx+C(y,z) $$

Now this should be my corresponding x component (if we look at it like a vector) Now to find y I derrived this in respect to y and I got this

$$ x^2-zx + Cy(y,z) = x^2-zx $$

We can cancel a few things out and are left with

$$ Cy(y,z) = 1 $$

Now how do I get my y (or z) constant from this.I have no y or z standing anywhere and I think I made a mistake somwhere.Am I doing something wrong or should I simply procced like I would usually (which is to integrate both sides)Thank you!