Vectors in Stokes and Gauss theorem

[Syrena]

Homework Statement

I have a couple of problems With normal vectors (this especially in Stokes theoerm where it get used often).

1) In some tasks they use unit normal vector, in some they use ordinary normal vector, is there any rules on when to use what? couse it seems pretty random, or dose it really matter.

2) There seem to be so many different ways to calculate a normal vector, depending on what surface we have (example from a sphere to a cylinder or parabolid). I can't seem to find anywhere where all these different ways to find different normal vectors are written down, cause I can't think for myself what kind of Equation to use.

Homework Equations

Normal vectors and normal unit vectors

The Attempt at a Solution

A lot of research on the web...

 

[Pythagorean]

The normal vector is just a vector, which means it has a magnitude and a unit vector. You can always decompose your vector into those two components. It's just like any other algebraic trick: sometimes it just makes the problem easier and it's hard to always generalize when that will be. It can be important when doing transforms (like from cartesian to polar) since polar coordinates don't have constant unit vectors.