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There is some issue about gradients that disturbs me, so I'd be glad if you could help me figure it out.

Say I have a scalar field [tex]\phi(\mathbf{r})[/tex], that is not yet known. What I know is a function that is the gradient of [tex]\phi[/tex], so that [tex]\mathbf{F}(\mathbf{r}) = \nabla\phi(\mathbf{r})[/tex]. I want to find [tex]\phi[/tex] from [tex]\mathbf{F}[/tex], ignoring the constants of course. What I thought of was:

[tex]d\phi = \sum_{i=1}^{3}\frac{\partial\phi}{\partial x_i} dx_i = \sum_{i=1}^{3} F_i dx_i[/tex]

And therefore:

[tex]\phi = \int d\phi = \int\sum_{i=1}^{3} F_i dx_i[/tex]

But if you try that with the two-dimensional example [tex]\phi = x^2 - xy[/tex], it doesn't work, and gives and gives [tex]x^2 - 2xy[/tex].

Can anyone please explain that to me?

Thanks!

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# Question about gradient

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