Calculating line integral of a vector field

Question

I already know how to calculate line integral of a vector field, but in a textbook (calculus by Ron Larson) i have seen equation $\int_C F.N ds$ ,which 'C' is integration path, 'F' is vector field and 'N' is unit normal vector of integration path and integration path has been represented by a vector valued function with parameter of arc length. So i have got a bit confused because i believe, instead of normal vector 'N', we should use unit tangent vector 'T'. How to Calculate line integral of a vector field when integration path has been represented by a vector valued function with parameter of arc length?

Answer 1

This is your typical line integral used for things like work: $$\int_C \mathbf F \cdot \, \mathrm d \mathbf r = \int_C \mathbf F \cdot \mathbf T \, \mathrm d s$$

Integrating the unit normal, however, gives the flux along a line: $$\int_C \mathbf F \cdot \mathbf N \, \mathrm d s$$