The Fundamental Theorem of Line Integralsand Green’s Theorem

Lucky Galvez

Institute of MathematicsUniversity of the Philippines

Diliman

Math 55 FTLI and Green’s Theorem

Recall: FTOC

Recall from Math 53:

Theorem

Let f(x) be a function that is continuous on [a, b]. Then∫ b

af(x) dx = F (b)− F (a)

where F ′(x) = f(x).

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

Math 55 FTLI and Green’s Theorem

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∫

C∇f · d~R = f(~R(b))− f(~R(a))

Proof. By the definition of line integrals,∫C

∇f · d~R =

∫ b

a

∇f(~R(t)) · ~R′(t) dt

=

∫ b

a

(∂f

∂x

dx

dt+∂f

∂y

dy

dt+∂f

∂z

dz

dt

)dt

=

∫ b

a

d

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

= f(~R(b))− f(~R(a)) (by FTOC) �

Math 55 FTLI and Green’s Theorem

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∫

C∇f · 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∫

C∇f · d~R = f(x2, y2, z2)− f(x1, y1, z1).

In other words, we evaluate f at the endpoints.

Math 55 FTLI and Green’s Theorem

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=∫C1

~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 is

independent of path.

Math 55 FTLI and Green’s Theorem

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 if∫C

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

Proof. Let

∫C

~F · d~R be independent of path and consider a

closed path C in D. We can regard C = C1 ∪ C2. Note that C1

and −C2 have the same initial and terminal points. Hence,∫C

~F · d~R =

∫C1

~F · d~R+

∫C2

~F · d~R

=

∫C1

~F · d~R−∫−C2

~F · d~R

= 0 �

Math 55 FTLI and Green’s Theorem

Fundamental Theorem of Line Integrals

We have seen that

∫C∇f · 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 .

Math 55 FTLI and Green’s Theorem

Fundamental Theorem of Line Integrals

The following follows immediately from the previous theorem.

Corollary

Let ~F be a conservative vector field with domain D. Then∫C

~F · d~R = 0

for any closed curve C in D.

Math 55 FTLI and Green’s Theorem

Fundamental Theorem of Line Integrals

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.

Math 55 FTLI and Green’s Theorem

Fundamental Theorem of Line Integrals

Example

Evaluate

∫C

~F · d~R where ~F (x, y) =⟨4x3y4 + 2x, 4x4y3 + 2y

⟩and C is given by ~R(t) =

⟨tπ cos t− 1, sin

(t2

)⟩, 0 ≤ t ≤ π.

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(π) = 〈−2, 1〉 .

Hence,

∫C

~F · d~R = f(−2, 1)− f(−1, 0) = 21− 1 = 20.

Math 55 FTLI and Green’s Theorem

Exercises

1 Given ~F (x, y) =

⟨y2

1 + x2, 2y tan−1 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) = 〈e−y,−xe−y〉 in moving an object from the pointA(0, 1) to the point B(2, 0).

Math 55 FTLI and Green’s Theorem

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

Math 55 FTLI and Green’s Theorem

Green’s 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− ∂P

∂y

)dA

Math 55 FTLI and Green’s Theorem

Remarks and Notation

1 The Green’s 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− ∂P

∂y

)dA =

∫∂D

P dx+Qdy

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

Math 55 FTLI and Green’s Theorem

Green’s Theorem

Green’s 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− ∂P

∂y

)dA =

∫∫D′

(∂Q

∂x− ∂P

∂y

)dA+

∫∫D′′

(∂Q

∂x− ∂P

∂y

)dA

=

∫∂D′

P dx+Qdy +

∫∂D′′

P dx+Qdy

Math 55 FTLI and Green’s Theorem

Green’s Theorem

Green’s 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).

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

(∂Q

∂x− ∂P

∂y

)dA =

∫C1

P dx+Qdy +

∫C2

P dx+Qdy

=

∫CP dx+Qdy

Math 55 FTLI and Green’s Theorem

Green’s 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, θ) : 1 ≤ r ≤ 2, 0 ≤ θ2π} .Hence, by Green’s Theorem,∫

C

x2y dx+ xy2 dy =

∫∫D

(∂(xy2)

∂x− ∂(x2y)

∂y

)dA

=

∫∫D

y2 − x2 dA

=

∫ 2π

0

∫ 2

1

r3(sin2 θ − cos2 θ) dr dθ = 0

Math 55 FTLI and Green’s Theorem

Exercises

1 Evaluate

∫C

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

from (0, 0) to (0, π) to (2, π) 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 + e√x) 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 Green’s 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 (π, 0) and the line segment from(π, 0) to (0, 0).

Math 55 FTLI and Green’s Theorem

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/

Math 55 FTLI and Green’s Theorem