Line integrals through a vector field:
Let F be a vector field. Define a smooth curve, C=r ( t ) :t ϵ [a ,b ], then the line integral
of F along C is:
❑ b
∫ F dr=∫ F (r ( t ))∙r ( t ) dt
'
C a
Fundamental theorem of calculus:
Let G be a continuous function on [a , b], and let g be a differentiable function such
that g' =G . Then:
b b
∫ G( x ) dx=∫ g '( x )dx=g ( b )−g(a)
a a
Conservative vector field:
A vector field, F , is a conservative vector field if there exists a scalar function, f ,
such that F is the gradient field of f (∇ f =F ).
Fundamental theorem of line integrals:
Let C be a smooth curve parameterised by r ( t ) , a ≤ t ≤ b, and let f , be a differentiable
function such that ∇ f =F is continuous on C . Then:
❑ ❑
∫ F dr=∫ ∇ f dr =f ( r ( b ) )−f (r ( a ) )
C c
Path independence:
If F is a continuous vector field with domain D , then the line integral is path
independent if:
❑ ❑
∫ F dr=∫ F dr
C1 C2
for any two paths C 1 and C 2 in D , that share initial and terminal points.
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