# Relationship between electric potential and electric field

## The Attempt at a Solution

I could find how to solve #2,4, but I don't understand what #1,3 need to me. How can I prove some functions are scalar field or vector field?

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How can I prove some functions are scalar field or vector field?
The first step would be to find out what scalar and vector fields are. Once you do that, the rest should be fairly obvious.

Scalar field means the function of points associating scalar value. Is it clear?Then should I do rotational transformation to prove?

I think you're trying to say that a scalar fields associates a scalar with each point in space, which is correct. So you have to show (show, not prove) that the potential does just that. You shouldn't need to go to the trouble of doing any transformations.

Don't we have to show the quantity of point is scalar or vector??

You can tell that by inspection. There is nothing but scalars in #1. In #3 the result is obviously a vector, since it is a gradient, which is a vector by definition.