Independence of Path and ConservativeVector Fields

MATH 311, Calculus III

J. Robert Buchanan

Department of Mathematics

Summer 2011

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Goal

We would like to know conditions on a vector field functionF(x , y) = 〈M(x , y), N(x , y)〉, so that if A and B are any twopoints fixed in the plane, then∫

CF(x , y) · dr

is the same regardless of the path C taken from A to B.

A

B

In this case the line integral is independent of path.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Goal

We would like to know conditions on a vector field functionF(x , y) = 〈M(x , y), N(x , y)〉, so that if A and B are any twopoints fixed in the plane, then∫

CF(x , y) · dr

is the same regardless of the path C taken from A to B.

A

B

In this case the line integral is independent of path.J. Robert Buchanan Independence of Path and Conservative Vector Fields

Connected Regions

Definition

A region D ⊂ R2 is called connected if every pair of points in Dcan be connected by a piecewise-smooth curve lying entirely inD.

Connected Not connected

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Simply Connected Regions

Definition

A region D ⊂ R2 is simply connected if every closed curve Cin D encloses only points in D

Simply connected Not simply connected

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Independence of Path (1 of 2)

TheoremSuppose that the vector field F(x , y) = 〈M(x , y), N(x , y)〉 iscontinuous on the open, connected region D ⊂ R2. Then theline integral

∫C F(x , y) · dr is independent of path if and only if F

is conservative on D.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Independence of Path (2 of 2)

Suppose the vector field is conservative.

F(x , y) = ∇f (x , y) =⟨

∂f∂x , ∂f

∂y

⟩= 〈M(x , y), N(x , y)〉

Select any two points A = (x0, y0) and B = (x1, y1) in Dand let C = (x(t), y(t)) for a ≤ t ≤ b be any pathconnecting A and B.

∫C

F(x , y) · dr =

∫C

M(x , y) dx + N(x , y) dy

=

∫ b

a

[∂f∂x

(x(t), y(t))x ′(t) +∂f∂y

(x(t), y(t))y ′(t)]

dt

=

∫ b

a

ddt

[f (x(t), y(t))] dt

= f (x1, y1)− f (x0, y0)

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Independence of Path (2 of 2)

Suppose the vector field is conservative.

F(x , y) = ∇f (x , y) =⟨

∂f∂x , ∂f

∂y

⟩= 〈M(x , y), N(x , y)〉

Select any two points A = (x0, y0) and B = (x1, y1) in Dand let C = (x(t), y(t)) for a ≤ t ≤ b be any pathconnecting A and B.

∫C

F(x , y) · dr =

∫C

M(x , y) dx + N(x , y) dy

=

∫ b

a

[∂f∂x

(x(t), y(t))x ′(t) +∂f∂y

(x(t), y(t))y ′(t)]

dt

=

∫ b

a

ddt

[f (x(t), y(t))] dt

= f (x1, y1)− f (x0, y0)

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Fundamental Theorem of Calculus for Line Integrals

TheoremSuppose that F(x , y) = 〈M(x , y), N(x , y)〉 is continuous in theopen, connected region D ⊂ R2 and that C is anypiecewise-smooth curve lying in D, with an initial point (x0, y0)and terminal point (x1, y1). Then, if F is conservative on D, withF = ∇f (x , y), we have∫

CF(x , y) · dr = f (x , y)

∣∣∣∣(x1,y1)

(x0,y0)

= f (x1, y1)− f (x0, y0).

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Example (1 of 4)

Suppose F(x , y) = (2xe2y + 4y3)i + (2x2e2y + 12xy2)j.1 Show that the line integral

∫C F(x , y) · dr is independent of

path.2 Evaluate the line integral along any piecewise-smooth

curve connecting (0, 1) to (2, 3).

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Example (2 of 4)

F(x , y) = (2xe2y + 4y3)i + (2x2e2y + 12xy2)j

1 If f (x , y) = x2e2y + 4xy3, then ∇f (x , y) = F(x , y) whichimplies F(x , y) is conservative and hence independent ofpath.

2 By the Fundamental Theorem of Calculus for Line Integrals∫C

F(x , y) · dr = f (x , y)

∣∣∣∣(2,3)

(0,1)

= 216 + 4e6.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Example (2 of 4)

F(x , y) = (2xe2y + 4y3)i + (2x2e2y + 12xy2)j

1 If f (x , y) = x2e2y + 4xy3, then ∇f (x , y) = F(x , y) whichimplies F(x , y) is conservative and hence independent ofpath.

2 By the Fundamental Theorem of Calculus for Line Integrals∫C

F(x , y) · dr = f (x , y)

∣∣∣∣(2,3)

(0,1)

= 216 + 4e6.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Example (3 of 4)

Suppose F(x , y) = x2y3i + x3y2j and then find the work donemoving from (0, 0) to (2, 1).

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Example (4 of 4)

F(x , y) = x2y3i + x3y2j

1 If f (x , y) =13

x3y3, then ∇f (x , y) = F(x , y) which implies

F(x , y) is conservative and hence independent of path.

2 By the Fundamental Theorem of Calculus for Line Integrals

W =

∫C

F(x , y) · dr = f (x , y)

∣∣∣∣(2,1)

(0,0)

=83.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Example (4 of 4)

F(x , y) = x2y3i + x3y2j

1 If f (x , y) =13

x3y3, then ∇f (x , y) = F(x , y) which implies

F(x , y) is conservative and hence independent of path.2 By the Fundamental Theorem of Calculus for Line Integrals

W =

∫C

F(x , y) · dr = f (x , y)

∣∣∣∣(2,1)

(0,0)

=83.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Closed Curves (1 of 4)

TheoremSuppose that F(x , y) is continuous in the open, connectedregion D ⊂ R2. Then F is conservative on D if and only if∫

C F(x , y) · dr = 0 for every piecewise-smooth closed curvelying in D.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Closed Curves (2 of 4)

Suppose∫

C F(x , y) · dr = 0 for every piecewise-smoothclosed curve lying in D.

Let P and Q be any two points in D.Let C1 and C2 be any two piecewise-smooth curvesconnecting P and Q as shown next.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Closed Curves (3 of 4)

P

Q

C1

C2

D

-2 -1 0 1 2

-2

-1

0

1

2

x

y

C = C1 ∪ (−C2) is a closed curve in D.J. Robert Buchanan Independence of Path and Conservative Vector Fields

Closed Curves (4 of 4)

0 =

∫C

F(x , y) · dr

=

∫C1

F(x , y) · dr +

∫−C2

F(x , y) · dr

=

∫C1

F(x , y) · dr−∫

C2

F(x , y) · dr∫C1

F(x , y) · dr =

∫C2

F(x , y) · dr

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Simply-connected Regions

Recall: a region is simply-connected if it contains no holes.

TheoremSuppose that M(x , y) and N(x , y) have continuous first partialderivatives on a simply-connected region D. Then∫

C M(x , y) dx + N(x , y) dy is independent of path in D if andonly if My (x , y) = Nx(x , y) for all (x , y) ∈ D.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Example

Show that the following line integral is independent of path.∫C

e2y dx + (1 + 2xe2y ) dy

Let M(x , y) = e2y and N(x , y) = 1 + 2xe2y . Then

My (x , y) = 2e2y = Nx(x , y)

and thus by the previous theorem the line integral isindependent of path.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Example

Show that the following line integral is independent of path.∫C

e2y dx + (1 + 2xe2y ) dy

Let M(x , y) = e2y and N(x , y) = 1 + 2xe2y . Then

My (x , y) = 2e2y = Nx(x , y)

and thus by the previous theorem the line integral isindependent of path.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Conservative Vector Fields

Summary: suppose that F(x , y) = 〈M(x , y), N(x , y)〉 whereM(x , y) and N(x , y) have continuous first partial derivatives onan open, simply-connected region D ⊂ R2. In this case thefollowing statements are equivalent.

1 F(x , y) is conservative in D.2 F(x , y) is a gradient field in D (i.e. F(x , y) = ∇f (x , y) for

some potential function f , for all (x , y) ∈ D.3∫

C F(x , y) · dr is independent of path in D.4∫

C F(x , y) · dr = 0 for every piecewise-smooth closedcurve C lying in D.

5 My (x , y) = Nx(x , y) for all (x , y) ∈ D.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Graphical Interpretation (1 of 2)

Suppose F(x , y) = 1√x2+y2

〈−y , x〉.

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

x

y

The vector field is not conservative since along the closed paththe vector field is always oriented with the curve (or alwaysagainst it).

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Graphical Interpretation (2 of 2)

Suppose F(x , y) = 〈y , x〉.

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

x

y

The vector field is conservative since along the closed path asmuch of the vector field is oriented with the curve as against it.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Three Dimensions (1 of 3)

Remark: much of what we have stated for two-dimensionalvector fields can be extended to three-dimensional vector fields.

ExampleEvaluate the line integral∫

Cy2 dx + (2xy + e3z) dy + 3ye3z dz

along any piecewise-smooth path connecting (0, 1, 1/2) to(1, 0, 2).

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Three Dimensions (2 of 3)

F(x , y , z) = 〈y2, 2xy + e3z , 3ye3z〉

1 If f (x , y , z) = xy2 + ye3z then F(x , y , z) = ∇f (x , y , z) whichimplies F(x , y , z) is conservative.

2 By the Fundamental Theorem of Calculus for Line Integrals∫C

F(x , y , z) · dr = xy2 + ye3z∣∣∣(1,0,2)

(0,1,1/2)= −e3/2.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Three Dimensions (2 of 3)

F(x , y , z) = 〈y2, 2xy + e3z , 3ye3z〉

1 If f (x , y , z) = xy2 + ye3z then F(x , y , z) = ∇f (x , y , z) whichimplies F(x , y , z) is conservative.

2 By the Fundamental Theorem of Calculus for Line Integrals∫C

F(x , y , z) · dr = xy2 + ye3z∣∣∣(1,0,2)

(0,1,1/2)= −e3/2.

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Three Dimensions (3 of 3)

TheoremSuppose that the vector field F(x , y , z) is continuous on theopen, connected region D ⊂ R3. Then, the line integral∫

C F(x , y , z) · dr is independent of path in D if and only if thevector field F is conservative in D, that is,F(x , y , z) = ∇f (x , y , z), for all (x , y , z) in D, for some scalarfunction f (a potential function for F). Further, for anypiecewise-smooth curve C lying in D, with initial point(x1, y1, z1) and terminal point (x2, y2, z2) we have∫

CF(x , y , z) · dr = f (x , y , z)

∣∣∣∣(x2,y2,z2)

(x1,y1,z1)

= f (x2, y2, z2)−f (x1, y1, z1).

J. Robert Buchanan Independence of Path and Conservative Vector Fields

Homework

Read Section 14.3.Exercises: 1–51 odd

J. Robert Buchanan Independence of Path and Conservative Vector Fields