15 ftli and greens thm - handout

The Fundamental Theorem of Line Integralsand Greens Theorem

Math 55 -Elementary Analysis III

Institute of MathematicsUniversity of the Philippines

Diliman

Math 55 FTLI and Greens Theorem 1/ 19

Recall: FTOC

Recall from Math 53:

Theorem

Let f(x) be a function that is continuous on [a, b]. Then baf(x) dx = F (b) F (a)

where F (x) = f(x).

Question: Is there an analog of this theorem for line integrals?


An analog of FTOC for Line Integrals

Theorem

Let C be a smooth curve given by a vector function ~R(t),a t b and f be a differentiable function whose gradientvector f is continuous on C. Then

Cf d~R = f(~R(b)) f(~R(a))

Proof. By the definition of line integrals,C

f d~R = ba

f(~R(t)) ~R(t) dt

=

ba

(f

x

dx

dt+f

y

dy

dt+f

z

dz

dt

)dt

=

ba

d

dtf(~R(t)) dt (by the Chain Rule)

= f(~R(b)) f(~R(a)) (by FTOC) Math 55 FTLI and Greens Theorem 3/ 19

Notation

If f is a function of two variables and C is a plane curve withinitial point A(x1, y1) and terminal point B(x2, y2), then

Cf d~R = f(x2, y2) f(x1, y1).

If f is a function of three variables and C is a plane curve withinitial point A(x1, y1, z1) and terminal point B(x2, y2, z2), then

Cf d~R = f(x2, y2, z2) f(x1, y1, z1).

In other words, we evaluate f at the endpoints.


Independence of Path

Suppose C1 and C2 are two curves (paths) having the sameinitial and terminal points. We know in general that

C1

~F d~R 6=C2

~F d~R

If the equality holds for any two paths C1 and C2 having the

same initial and terminal points, then we say that

C

~F d~R isindependent of path.


Independence of Path

Theorem

Let ~F be a continuous vector field with domain D. The line

integral

C

~F d~R is independent of path in D if and only ifC

~F d~R = 0 for every closed path C in D.

Proof. Let

C

~F d~R be independent of path and consider aclosed path C in D. We can regard C = C1 C2. Note that C1and C2 have the same initial and terminal points. Hence,

C

~F d~R =C1

~F d~R+C2 ~F d~R

=

C1

~F d~RC2

~F d~R= 0


Fundamental Theorem of Line Integrals

We have seen that

Cf d~R is independent of path.

If we let ~F = f , i.e., ~F is a conservative vector field withpotential function f , then we have the following theorem:

Theorem (FTLI)

The line integral of a conservative vector field ~F is independentof path. That is, if C is a smooth curve given by ~R(t), a t bwith initial point A(x1, y1, z1) and terminal point B(x2, y2, z2)and ~F is a conservative vector field which is continuous on C,then

C

~F d~R = f(x2, y2, z2) f(x1, y1, z1)

where f is a potential function for ~F .



The following follows immediately from the previous theorem.

Corollary

Let ~F be a conservative vector field with domain D. ThenC

~F d~R = 0

for any closed curve C in D.



Example

Show that the vector field ~F (x, y) =y2, 2xy

is conservative

and evaluate

C

~F d~R where C is the unit circle.

Solution. Recall that ~F = P,Q is conservative iff Py = Qx.Since

yy2 = 2y =

x2xy,

F (x, y) =y2, 2xy

is conservative.

Now, C is a closed curve so by the previous corollary,C

~F d~R = 0.



Example

Evaluate

C

~F d~R where ~F (x, y) = 4x3y4 + 2x, 4x4y3 + 2yand C is given by ~R(t) =

tpi cos t 1, sin

(t2

), 0 t pi.

Solution. We can use the definition of line integrals but thatwould give a nasty solution.Instead, we can show that ~F is conservative with potentialfunction

f(x, y) = x4y4 + x2 + y2 + c.

To use FTLI, we find the endpoints of C. Note that

~R(0) = 1, 0 and ~R(pi) = 2, 1 .

Hence,

C

~F d~R = f(2, 1) f(1, 0) = 21 1 = 20.Math 55 FTLI and Greens Theorem 10/ 19

Exercises

1 Given ~F (x, y) =

y2

1 + x2, 2y tan1 x

,

a. Show that ~F is conservative and find a potential function.

b. Evaluate

C

~F d~R for any path C from (0, 0) to (1, 2).

2 Evaluate

C

sin y dx+ x cos y dy, where C is the ellipse

x2 + xy + y2 = 1.

3 Find the work done by the force field~F (x, y) = ey,xey in moving an object from the pointA(0, 1) to the point B(2, 0).


Orientation of a Curve

Let C be a simple closed curve. The positive orientation ofC refers to the single counterclockwise traversal of C.

Figure: positive orienation Figure: negative orienation


Greens Theorem

Theorem

Let C be a positively oriented, piecewise-smooth, simple closedcurve in the plane and let D be the region bounded by C. If Pand Q have continuous partial derivatives on an open regionthat contains D, then

CP dx+Qdy =

D

(Q

x Py

)dA


Remarks and Notation

1 The Greens Theorem relates a line integral along a curveC and the double integral over the plane region D boundedby C.

2 The notation

CP dx+Qdy is sometimes used to indicate

that the line integral is calculated using the positiveorientation of C.

3 Other texts denote the positively oriented boundary curveof D as D and hence

D

(Q

x Py

)dA =

D

P dx+Qdy

4 The Greens Theorem can be regarded, in some sense, asthe analog of FTOC for double integrals.


Greens Theorem

Greens Theorem will also hold if the region D is not simplyconnected, i.e, the boundary of D is C = C1 C2 (assume theyare positively oriented).

D

(Q

x Py

)dA =

D

(Q

x Py

)dA+

D

(Q

x Py

)dA

=

D

P dx+Qdy +

D

P dx+Qdy


Greens Theorem

The line integrals along each of the common boundary pointsare on opposite directions, hence they cancel and we getD

(Q

x Py

)dA =

C1

P dx+Qdy +

C2

P dx+Qdy

=

CP dx+Qdy


Greens Theorem

Example

Evaluate

Cx2y dx+ xy2 dy where C is the boundary of the

region D between the circles x2 + y2 = 4 and x2 + y2 = 1.

Solution. Note that the region D can be expressedconveniently in polar coordinates, i.e.,

D = {(r, ) : 1 r 2, 0 2pi} .Hence, by Greens Theorem,

C

x2y dx+ xy2 dy =

D

((xy2)

x (x

2y)

y

)dA

=

D

y2 x2 dA

=

2pi0

21

r3(sin2 cos2 ) dr d = 0Math 55 FTLI and Greens Theorem 17/ 19

Exercises

1 Evaluate

C

cos y dx+ x2 sin y dy where C is the rectangle

from (0, 0) to (0, pi) to (2, pi) to (2, 0) to (0, 0).

2 Evaluate

Cx2y dx+ xy2 dy where C is the positively

oriented triangle with vertices at (0, 0), (1, 0) and (0, 1).

3 Evaluate

C

(y + ex) dx+ (2x+ cos y2) dy, where C is the

positively oriented boundary of the region enclosed by theparabolas y = x2 and x = y2.

4 Use Greens Theorem to evaluate

C

~F d~R where~F (x, y) =

x+ y3, x2 +

y

and C consists of the curvey = sinx from (0, 0) to (pi, 0) and the line segment from(pi, 0) to (0, 0).


References

1 Stewart, J., Calculus, Early Transcendentals, 6 ed., ThomsonBrooks/Cole, 2008

2 Dawkins, P., Calculus 3, online notes available athttp://tutorial.math.lamar.edu/


15 ftli and greens thm - handout

Documents