Path (English)
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Definition (keystage 3)
A route from one place to another. Sometimes we integrate, or perform some calculation, along a path.
Definition (undergraduate level)
A path from x to y in a topological space X is a continuous function ϕ: [a,b] → X such that ϕ(a)=x and ϕ(b)=y .
X is path connected if there is such a path for any x,y ∈ X .
A path is smooth if it is continuously differentiable.
A path is piecewise smooth if it is the join of finitely many smooth paths, where the join of two paths is the path formed by adjoining two paths where the endpoint of one is the starting point of the other, with a linear change of variable in the parameter as necessary. If each of the paths joined is linear then the resulting join is a polygonal path.
X is polygonally connected if there is such a path for any x,y ∈ X .
For subsets of Rn , path connectedness, polygonal connectedness and connectedness are all equivalent.
Finally, the reverse of ϕ is the path − ϕ(t) : [−b,−a] → X , − ϕ(t) = ϕ(−t) .
There is an equivalence relation on X defined by x ∼ y if there is a continuous path from x to y . The equivalence classes are called the path components of X .
A path is closed if ϕ(a) = ϕ(b) ; it is simple if it is injective, possibly (in the case of a closed path) except for the endpoints.
Two paths ϕ: [a,b] → X and ψ: [c,d] → X are said to be equivalent if there is a continuously differentiable function f : [a,b] → [c,d] such that f′(t) > 0 and ϕ(t) = ψ(f(t)) for all t ∈ [a,b] , and f(a)=c , f(b)=d . This defines an equivalence relation on the set of possible paths in X , and the members of the equivalence classes all have the same track, or image, in X .
We define the integral along a path ϕ of a continuous complex function f defined on a domain within which the track of ϕ lies as
if ϕ is smooth, where a and b are the parameter points corresponding to the endpoints of ϕ . If ϕ is merely piecewise smooth, we split it into smooth parts and sum the integrals.
See also length.
X is path connected if there is such a path for any x,y ∈ X .
A path is smooth if it is continuously differentiable.
A path is piecewise smooth if it is the join of finitely many smooth paths, where the join of two paths is the path formed by adjoining two paths where the endpoint of one is the starting point of the other, with a linear change of variable in the parameter as necessary. If each of the paths joined is linear then the resulting join is a polygonal path.
X is polygonally connected if there is such a path for any x,y ∈ X .
For subsets of Rn , path connectedness, polygonal connectedness and connectedness are all equivalent.
Finally, the reverse of ϕ is the path − ϕ(t) : [−b,−a] → X , − ϕ(t) = ϕ(−t) .
There is an equivalence relation on X defined by x ∼ y if there is a continuous path from x to y . The equivalence classes are called the path components of X .
A path is closed if ϕ(a) = ϕ(b) ; it is simple if it is injective, possibly (in the case of a closed path) except for the endpoints.
Two paths ϕ: [a,b] → X and ψ: [c,d] → X are said to be equivalent if there is a continuously differentiable function f : [a,b] → [c,d] such that f′(t) > 0 and ϕ(t) = ψ(f(t)) for all t ∈ [a,b] , and f(a)=c , f(b)=d . This defines an equivalence relation on the set of possible paths in X , and the members of the equivalence classes all have the same track, or image, in X .
We define the integral along a path ϕ of a continuous complex function f defined on a domain within which the track of ϕ lies as
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See also length.
Example (undergraduate level)
Sometimes we integrate, or perform some calculation, along a path.
Relations
- broader:
- (en) Curve
- narrower:
- (en) Circuit
- (en) Geodesic
- (en) Trajectory
- referenced:
- (en) Equivalent
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