How to prove some functions are scalar field or vector field

[Flotensia]

Homework Statement

Homework Equations

The Attempt at a Solution

I solved #2,4 but I don't understand what #1,3 need to me. I know that scalar field is a function of points associating scalar value. But how can I prove some function is scalar field or vector field?

 

[vela]

That's a good question. What does your textbook say?

 

[Flotensia]

Text says scalar is invariant in rotational transformation.Then to prove some functions are scalar field, should I do rotational transformation, or are there any other methods?