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In the composition:

$F = -\nabla \phi + \nabla \times A$

Why is there a negative in front of the gradient field? Is this always required?

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    It's a convention. That's all. It's introduced because in some applications, $\phi$ is a potential. Let $\psi=-\phi$ and you have eliminated the minus sign. – 2017-02-02
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    Just convention, really. It's not required - everything works just fine without it, as you could just define it to be a part of $\phi$. The reason it's there comes from Physics, where the force $F$ is the negative gradient of the potential $\phi$. – 2017-02-02
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    @tom Great minds and all .... – 2017-02-02

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