How and where to use Static modifier in Java?

Question

How and where should we use a Static modifier for:

1. Field and
2. Method?

For example in java.lang.Math class, the fields

Answer 1

Usually when the method only depends on the function parameters and not on the internal state of the object it's a static method (with singletone being the only exception). I can't imagine where static fields are really used (they're the same as global variables which should be avoided).
Like in your example the math functions only depend on the parameters.