Confusion in Stoke's theorem!

  • #1
270
1

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!

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The Attempt at a Solution

 

Answers and Replies

  • #2
Defennder
Homework Helper
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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.
 
  • #3
270
1
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?
 
  • #4
Defennder
Homework Helper
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Eg.
A scalar function of 3 variables gives you a single (scalar) value [tex]f(x,y,z) = x + y + z[/tex] for every point (x,y,z).

A vector function of 3 variables gives you a vector: [tex]\mathbf{F}(x,y,z) = x \mathbf{i} + y\mathbf{j} + z\mathbf{k}[/tex] for every point (x,y,z)
 
  • #5
270
1
oh that makes sense thanks the other definitions I found online I couldn't really understand but this clears it up well.
 

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