Line integrals:
If f is defined over a smooth curve, C , then the line integral of f along C is:
❑
∫ f ( x , y ) ds=lim
n →∞
¿ ¿
f (x i , y i )∆ s i
C
More commonly, it is defined as:
√(
b b
)( )
❑
dx 2 dy 2
∫ f ( x , y ) ds=∫ f ( r ( t ) ) dt ¿ ∫ f ( r ( t ) ) ∙‖r ( t )‖dt
'
+
C a dt dt a
The value of the integral does not depend on the parameterisation of the curve,
provided that the curve is traversed exactly once as t increases from a to b .
Piecewise-smooth curves:
Suppose C is a piecewise-smooth curve so that:
C=C 1 ∪C 2 … ∪ C n
where the initial point, C i+1 is the terminal point of C i. Then:
❑ ❑ ❑ ❑
∫ f ( x , y ) ds=∫ f ( x , y ) ds+∫ f ( x , y ) ds+ …+∫ f ( x , y ) ds
C C1 C2 Cn
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