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If we are given a vector function, can we directly write its associated scalar potential? Should there be some other "cross" terms too?

Let's take a Vector function $$\vec A(x, y, z) = (4xy-3x^2z^2)\hat{i} + 2x^2\hat{j} + 2x^3z \ \hat k$$
I chose this specifically because this leads to some error.

And now I try to write scalar potential by writing $$\frac{\partial\phi_1}{\partial x} = 4xy-3x^2z^2$$ and integrating to get $$\phi_1 = 2x^2y-x^3z^2 + C_1$$ i.e. the scalar function from x part.

Similarly doing for $y$ and $z$ also.

Now final scalar potential will be $$\phi_1 + \phi_2 + \phi_3 + C$$ with a final reduced constant. It will be $$\phi = 4xy-x^3z^2 -2x^3z + C$$

But when I try to get $\vec A$ again by taking the gradient of this function, it comes with some other terms too. Whats wrong here

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    $\begingroup$ Can a force vector always be written in terms of a potential, or does it need to satisfy certain conditions? $\endgroup$
    – Oбжорoв
    Commented Dec 25, 2020 at 10:48
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    $\begingroup$ Please edit your question to use Mathjax which is the site standard for math expressions. $\endgroup$
    – StephenG - Help Ukraine
    Commented Dec 25, 2020 at 11:01
  • $\begingroup$ Hi Shikhar Chamoli. Welcome to Phys.SE. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. $\endgroup$
    – Qmechanic ♦
    Commented Dec 25, 2020 at 11:39
  • $\begingroup$ @ShikharChamoli The edit history of your post clearly shows that your original question stated the problem as "\vectorA = (4xy-3x²z²)î + 2x²j + (2x³z)k.". In itself, this is not too serious, people make mistakes, but I see no point in drawing this out. And I certainly see no point in attacking Qmechanic unnecessarily, given the monumental amount of work they do for this site with little thanks. $\endgroup$
    – Philip
    Commented Dec 25, 2020 at 11:46
  • $\begingroup$ Your question is not asking a question and giving an example, you're asking what is going wrong in your particular attempt to determine the scalar potential for this particular vector field. As we've seen, a simple + vs. - typo in the example can change the meaning of your question, since the answer you have already received becomes incorrect for the version with '-'. $\endgroup$
    – ACuriousMind ♦
    Commented Dec 25, 2020 at 11:55

1 Answer 1

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No it needs to be conservative field. In three dimensions, this means the vector field must have 0 curl. Your example has curl $-(6x^2 + 4 xz )\hat{j}$.

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  • $\begingroup$ Sorry @Bobak Hashemi sir, it was minus originally in the k components $\endgroup$
    – Shikhar Chamoli
    Commented Dec 25, 2020 at 11:49
  • $\begingroup$ @ShikharChamoli you don't just add up the terms, each of the $\phi_1$, $\phi_2$, etc.. give necessary terms, its up to you to put them together into a total function $\phi$ that includes all necessary terms. The final answer (assuming you reverse the sign of the z term is) is $\phi = 2x^2 y - x^3 z^2 + C$ $\endgroup$
    – Something Different
    Commented Dec 25, 2020 at 23:23

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