Fundamental Theorem of Line Integrals Lesson 16.3

University: Santa Monica College

Course: Multivariable Calculus (MATH 11)

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Calc 3 Handout Section 16.3: Fundamental Theorem of Line Integrals (FTLI or FToLI) I. Lai

Recall from the previous section that the work done by a force field along a curve was given by

Work Integral

C C

F dr P dx Q dy÷= +

òò

in 2-space where

( ) ( )

, , ,F P x y Q x y=

C C

F dr P dx Q dy R dz÷= + +

òò

in 3-space where

( ) ( ) ( )

, , , , , , , ,F P x y z Q x y z R x y z=

The parametric curve C in a work integral is called the path of integration. One important problem in

applications is to determine how the path of integration affects the work performed by a force field on a

particle that moves from a fixed point A to a fixed point B.

We will show that if the force field

F

is conservative, then the work that the field performs on a particle

does not depend on (is independent of) the particular path C that the particle follows.

But before we can make that connection, we have to first talk about the FTLI (or FToLI).

Theorem (Fundamental Theorem of Line Integrals or FToLI)

Let C be a smooth curve given by the vector function

( )r t

,

a t bóó

. Let

f

be a differentiable function

of two or three variables whose gradient vector

fñ

is continuous on C. Then

( ) ( )

( ) ( )

C

f dr f r b f r añ÷ = 2

ò

Proof:

This says that we can evaluate the line integral of a gradient vector field simply by knowing the value of

f

at the endpoints of C. In fact, this means:

( ) ( )

2 2 1 1

, ,

C

f dr f x y f x yñ÷ = 2

ò

for a plane curve

( ),r t a t bóó

, with initial point

( )

1 1

( ) ,r a x y=

and

terminal point

( )

2 2

( ) ,r b x y=

( ) ( )

2 2 2 1 1 1

, , , ,

C

f dr f x y z f x y zñ÷ = 2

ò

for a space curve

( ),r t a t bóó

with initial point

( )

1 1 1

( ) , ,r a x y z=

and end point

( )

2 2 2

( ) , ,r b x y z=

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