Lecture Notes 7: Integration along Paths

Ruipeng Shen

May 29, 2014

1 Paths

Definition 1. A path c(t) : [a, b] Rn is called C1 if and only if its component functions havecontinuous derivatives on [a, b]. A path c is a piecewise C1 path if its domain [a, b] can bebroken into subintervals

[a, b] = [t0, t1] [t1, t2] [ti1, ti] [tn1, tn]

so that the restriction of c to each subinterval [ti1, ti] is a C1 path.

Definition 2. A curve with the same initial and terminal point is called a closed curve.

Definition 3. (a) The image of a one-to-one, piecewise C1 path c : [a, b] Rm is called asimple curve.(b) The image of a piecewise C1 path c : [a, b] Rm that is one-to-one on [a, b) and is such thatc(a) = c(b) is called a simple closed curve.

Example 4. The formula c(t) = (sin t, sin 2t) gives

a simple curve if t [0, pi/2]. a simple closed curve if t [0, pi]. a closed (but not simple) if t [0, 2pi].

2 Path Integrals of a Real-valued Function

Fence Model A path integral of a real-valued function f over a C1 path c(t) could be under-stood as the area of a fence erected along the path of height f(c(t)) at c(t).

Definition 5. Let c : [a, b] R2 be a C1 path and f : R2 R be a function such that thecomposition f(c(t)) is continuous on [a, b]. The path integral

cfds of f along c is given by

c

fds =

ba

f(c(t))c(t)dt = ba

f(x(t), y(t))

(x(t))2 + (y(t))2dt.

If c is a piecewise C1 path consisting of C1 paths cj, j = 1, 2, ,m, thenc

fds =

mj=1

cj

fds.

Example 6. What is the integral of function 1 over a C1 path ?

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Example 7. Find the following integral over the parabola c(t) = (2t, t2), t [0, 1]c

xds.

Example 8. Find the following integral over the helix c(t) = (cos t, sin t,

3t), t [0, 2pi]c

(xy + z)ds.

Theorem 9. Let c be a C1 path in Rm and = c be a reparametrization of c, thenc

fds =

fds.

3 Path Integrals of Vector Fields

Work Model Suppose that the vector field F(x) represents a force on an object, then thepath integral of F along a path c can be understood as the work done by the force if the objectmoves along curve from the initial point to the terminal point. By taking a partition

a = t0 < t1 < t2 < < tn1 < tn = b; ti = ti ti1We have

W =

ni=1

Wi ni=1

F(c(ti1)) (c(ti) c(ti1))

ni=1

F(c(ti1)) c(ti1)ti ba

F(c(t)) c(t)dt.

Definition 10. Let c(t) : [a, b] Rm be a C1 path and let F : Rm Rm be a vector fieldsuch that the composition F(c(t)) is continuous on [a, b]. The path integral (or the line integral)cF ds of F along c is defined by

c

F ds = ba

F(c(t)) c(t)dt.

Alternative Notation If F(x, y) = (F1(x, y), F2(x, y)), then we define (Assume c(t) = (x(t), y(t)))c

F1(x, y)dx+ F2(x, y)dy =

c

F ds = ba

[F1(x(t), y(t))x(t) + F2(x(t), y(t))y(t)] dt.

Example 11. Compute the path integral of F(x, y) = (ex+y, 3x) along the path c(t) = (t2, 32t2) with t [1, 1].Example 12. Compute the path integral of a constant F0 along an arbitrary path c.

Theorem 13. Let F be a continuous vector field on Rm. Let c(t1) : [a, b] Rm be a C1 curveand let = c((t2)) be a reparametrization of c. Then

F ds ={

cF ds If is orientation-preserving;

cF ds If is orientation-reversing;

Proposition 14. If a path c consists of cj with j = 1, 2, , n; thenc

F ds =nj=1

cj

F ds.

Example 15. The line integral of a vector field along an oriented simple closed curve does notdepend on the initial point.

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4 Path Integrals Independent of Path

Example 16. Find the line integral of gravitational field F(x) =GMmxx3 along an arbitrary

path c(t).

Definition 17. A continuous vector field F : U Rm is called a gradient vector field if andonly if there is a differentiable function f : U R such that F = f .Example 18. The gravitational field is given by F = V . Here V is the gravitational potentialV = GMm/x. In general

c

f ds = f(c(b)) f(c(a)).

Definition 19. We say a set U Rm is connected if any two points in U can be joined by acontinuous curve that is completely contained in U .

Theorem 20. If F is a continuous vector field defined on an open connected domain U , thenthe following are equivalent to each other

For any oriented, simple closed curve c, cF ds = 0.

F is path-independent, for any two oriented, simple curve c1 and c2 having the same initialand terminal point points,

c1

F ds = c2

F ds. F is a gradient vector field.

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