path integral

path integral

The

path integral

is a generalization of the

integral

that is very useful in theoretical and applied physics. Consider a

vector field

F

→

:

ℝ

n

→

ℝ

m

and a path (

http://planetmath.org/PathConnected

)

γ

⊂

ℝ

n

. The path integral of

F

→

along the path

γ

is defined as a

definite integral

. It can be constructed to be the

Riemann sum

of the values of

F

→

along the curve

γ

. Thusly, it is defined in terms of the parametrization of

γ

, mapped into the domain

ℝ

n

of

F

→

. Analytically,

∫

γ

F

→

⋅

𝑑

x

→

=

∫

a

b

F

→

⁢

(

γ

→

⁢

(

t

)

)

⋅

𝑑

x

→

where

γ

→

⁢

(

a

)

,

γ

→

⁢

(

b

)

are elements of

ℝ

n

, and

d

⁢

x

→

=

⟨

d

⁢

x

1

d

⁢

t

,

⋯

,

d

⁢

x

n

d

⁢

t

⟩

⁢

d

⁢

t

where each

x

i

is parametrized into a function of

t

.

Proof and existence of path integral:

Assume we have a

parametrized curve

γ

→

⁢

(

t

)

with

t

∈

[

a

,

b

]

. We want to construct a sum of

F

→

over this

interval

on the curve

γ

. Split the interval

[

a

,

b

]

into

n

subintervals

of size

Δ

⁢

t

=

(

b

-

a

)

/

n

. Note that the

arc lengths

need not be of equal length, though the intervals are of equal size. Let

t

i

be an element of the

i

th subinterval. The quantity

|

γ

→

′

⁢

(

t

i

)

|

gives the

average

magnitude of the vector

tangent

to the curve at a point in the interval

Δ

⁢

t

.

|

γ

→

′

⁢

(

t

i

)

|

⁢

Δ

⁢

t

is then the approximate arc length of the curve segment produced by the subinterval

Δ

⁢

t

. Since we want to sum

F

→

over our curve

γ

→

, we let the range of our curve equal the domain of

F

→

. We can then dot this vector with our

tangent vector

to get the approximation to

F

→

at the point

γ

→

⁢

(

t

i

)

. Thus, to get the sum we want, we can take the limit as

Δ

⁢

t

approaches 0.

lim

Δ

⁢

t

→

0

⁡

∑

a

b

F

→

⁢

(

γ

→

⁢

(

t

i

)

)

⋅

γ

→

′

⁢

(

t

i

)

⁢

Δ

⁢

t

This is a Riemann sum, and thus we can write it in integral form. This integral is known as a path or

line integral

(the older name).

∫

γ

F

→

⋅

𝑑

x

→

=

∫

a

b

F

→

⁢

(

γ

→

⁢

(

t

)

)

⋅

γ

→

′

⁢

(

t

)

⁢

𝑑

t

Note that the path integral only exists if the definite integral exists on the interval

[

a

,

b

]

.

Properties

:

A path integral that begins and ends at the same point is called a closed path integral, and is denoted with the summa symbol with a centered circle:

∮

. These types of path integrals can also be evaluated using Green’s

theorem

.

Another property of path integrals is that the directed path integral on a path

Γ

in a vector field is equal to the

negative

of the path integral in the opposite

direction

along the same path. A directed path integral on a closed path is denoted by summa and a circle with an arrow denoting direction.

Visualization Aids:

This is an image of a path

γ

superimposed on a vector field

F

→

.

This is a visualization of what we are doing when we take the integral under the curve

S

:

P

→

F

→

.

Title

path integral

Canonical

name

PathIntegral

Date of creation

2013-03-22 12:16:14

Last modified on

2013-03-22 12:16:14

Owner

slider142 (78)

Last modified by

slider142 (78)

Numerical id

19

Author

slider142 (78)

Entry type

Definition

Classification

msc 81S40

Classification

msc 46T12

Synonym

line integral

Related topic

ComplexIntegral

Related topic

ContourIntegral

Related topic

RealAndImaginaryPartsOfContourIntegral

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