Relationship between electric potential and electric field

AI Thread Summary
To determine if functions are scalar or vector fields, one must understand the definitions of each. A scalar field associates a scalar value with each point in space, while a vector field associates a vector with each point. For the given problems, it is sufficient to inspect the functions: if they consist solely of scalars, they are scalar fields, and if they involve gradients, they are vector fields. Transformations are unnecessary for this classification. The key is to recognize the nature of the quantities involved in each problem.
Flotensia
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Homework Statement


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Homework Equations

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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Flotensia said:
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.
 
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