Line integral

A line integral of a vector function f along a line or curve segment C is the following (all three formulations are equivalent);

$\int _{\mathbf {C} }{\vec {f}}({\vec {s}})\cdot d{\vec {s}}$

$\int _{\mathbf {C} }f_{1}s_{1}dx+f_{2}s_{2}dy+f_{3}s_{3}dz$

$\int _{\mathbf {C} }\langle {\vec {f}}({\vec {s}}),d{\vec {s}}\rangle$