Understanding the Confusion in Stoke's Theorem Formulas

[physstudent1]

Homework Statement

Hello, this isn't a specific problem but a part I am confused about in Stoke's theorem. In my text the section on Line integrals (if C(t) is the parameterization of the curve) as the integral of F(C(t))*C'(t) but for vector fields the formula becomes the integral of F(C(t))*||C'(T)|| now I understand this but when I got to Stoke's theorem it says The line integral of the boundary of a surface is equal to the surface integral of the curl vector. However In every example I have seen in my text the F is a vector field so I figured that the line integral should be defined as the integral of F(C(t))*||C'(T)|| but it is the integral of F(C(t))*C'(t) can anyone clear this up for me please if you don't understand my question please ask I will try to reword it more clearly! Thanks!

Homework Equations

The Attempt at a Solution

 

[Defennder]

Hello, this isn't a specific problem but a part I am confused about in Stoke's theorem. In my text the section on Line integrals (if C(t) is the parameterization of the curve) as the integral of F(C(t))*C'(t)

Ok so this is for general line integrals of vector functions.

but for vector fields the formula becomes the integral of F(C(t))*||C'(T)||

This is the formula for line integrals of scalar fields, not vector fields.

However In every example I have seen in my text the F is a vector field so I figured that the line integral should be defined as the integral of F(C(t))*||C'(T)|| but it is the integral of F(C(t))*C'(t) can anyone clear this up for me please if you don't understand my question please ask I will try to reword it more clearly! Thanks!

As above.

 

[physstudent1]

thanks a lot for clearing this up

one more thing, just exactly how could I tell the differnece between a scalar function and a vector function?

 

[Defennder]

Eg.
A scalar function of 3 variables gives you a single (scalar) value $$f(x,y,z) = x + y + z$$ for every point (x,y,z).

A vector function of 3 variables gives you a vector: $$\mathbf{F}(x,y,z) = x \mathbf{i} + y\mathbf{j} + z\mathbf{k}$$ for every point (x,y,z)

 

[physstudent1]

oh that makes sense thanks the other definitions I found online I couldn't really understand but this clears it up well.