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4.4: Path Independence - Mathematics


We say the integral (int_{gamma} f(z) dz) is path in dependent if it has the same value for any two paths with the same endpoints. More precisely, if (f(z)) is defined on a region (A) then (int_{gamma} f(z) dz) is path independent in (A), if it has the same value for any two paths in (A) with the same endpoints.

The following theorem follows directly from the fundamental theorem. The proof uses the same argument as Example 4.3.2.

Theorem (PageIndex{1})

If (f(z)) has an antiderivative in an open region (A), then the path integral (displaystyle int_{gamma} f(z) dz) is path independent for all paths in (A).

Proof

Since (f(z)) has an antiderivative of (f(z)), the fundamental theorem tells us that the integral only depends on the endpoints of (gamma), i.e.

[int_{gamma} f(z) dz = F(z_1) - F(z_0) onumber]

where (z_0) and (z_1) are the beginning and end point of (gamma).

An alternative way to express path independence uses closed paths.

Theorem (PageIndex{2})

The following two things are equivalent.

  1. The integral (displaystyle int_{gamma} f(z) dz) is path independent.
  2. The integral (displaystyle int_{gamma} f(z) dz) around any closed path is 0.
Proof

This is essentially identical to the equivalent multivariable proof. We have to show two things:

  1. Path independence implies the line integral around any closed path is 0.
  2. If the line integral around all closed paths is 0 then we have path independence.

To see ((i)), assume path independence and consider the closed path (C) shows in figure (i) below. Since the starting point (z_0) in the same as the endpoint (z_1) the integral (int_C f(z) dz) must have the same value as the line integral over the curve consisting of the single point (z_0). Since that is clearly 0 we must have the integral over (C) is 0.

To see ((ii)), assume (int_C f(z) dz = 0) for any closed curve. Consider the two curves (C_1) and (C_2) shown in figure (ii). Both start at (z_0) and end at (z_1). By the assumption that integrals over closed paths are 0 we have (int_{C_1 - C_2} f(z) dz = 0). So,

[f_{C_1} f(z) dz = int_{C_2} f(z) dz. onumber]

That is, any two paths from (z_0) to (z_1) have the same line integral. This shows that the line integrals are path independent.


Independence Number

The (upper) vertex independence number of a graph, often called simply "the" independence number, is the cardinality of the largest independent vertex set, i.e., the size of a maximum independent vertex set (which is the same as the size of a largest maximal independent vertex set). The independence number is most commonly denoted , but may also be written (e.g., Burger et al. 1997) or (e.g., Bollobás 1981).

The independence number of a graph is equal to the largest exponent in the graph's independence polynomial.

The lower independence number may be similarly defined as the size of a smallest maximal independent vertex set in (Burger et al. 1997).

The ratio of the independence number of a graph to its vertex count is known as the independence ratio of (Bollobás 1981).

(A. E. Brouwer, pers. comm., Dec. 17, 2012).

The matching number of a graph is equal to the independence number of its line graph .

where is the vertex cover number of and its vertex count (West 2000).

Known value for some classes of graph are summarized below.

graph OEISvalues
alternating group graph A0000001, 1, 4, 20, 120, .
-Andrásfai graph () A0000273, 4, 5, 6, 7, 8, 9, 10, 11, 12, .
-antiprism graph () A0045232, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, .
-Apollonian network A0002441, 3, 9, 27, 81, 243, 729, 2187, .
complete bipartite graph A0000271, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, .
complete graph 1A0000121, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, .
complete tripartite graph A0000271, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, .
cycle graph () A0045261, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, .
empty graph A0000271, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, .
-folded cube graph () A0586221, 1, 4, 5, 16, 22, 64, 93, 256, .
grid graph A0009821, 2, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, .
grid graph A0364861, 4, 14, 32, 63, 108, 172, 256, 365, 500, .
-halved cube graphA0058641, 1, 4, 5, 16, 22, 64, 93, 256, .
-Hanoi graph A0002441, 3, 9, 27, 81, 243, 729, 2187, .
hypercube graph A0000791, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, .
-Keller graph A2589354, 5, 8, 16, 32, 64, 128, 256, 512, .
-king graph () A0087941, 4, 4, 9, 9, 16, 16, 25, 25
-knight graph () A0309784, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, .
Kneser graph
-Mycielski graph A2665501, 1, 2, 5, 11, 23, 47, 95, 191, 383, 767, .
Möbius ladder () A1096133, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, .
odd graph A0000001, 1, 4, 15, 56, 210, 792, 3003, 11440, .
-pan graph A0000002, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, .
path graph A0045261, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, .
prism graph () A0529282, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, .
-Sierpiński carpet graph4, 32, 256, .
-Sierpiński sieve graph1, 3, 6, 15, 42, .
star graph A0283101, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, .
triangular graph () A0045261, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, .
-web graph () A0327664, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, .
wheel graph A0045261, 2, 2, 3, 3, 4, 4, 5, 5, .

Precomputed independence numbers for many named graphs can be obtained in the Wolfram Language using GraphData[graph, "IndependenceNumber"].

Bollobás, B. "The Independence Ratio of Regular Graphs." Proc. Amer. Math. Soc. 83, 433-436, 1981.

Burger, A. P. Cockayne, E. J. and Mynhardt, C. M. "Domination and Irredundance in the Queens' Graph." Disc. Math. 163, 47-66, 1997.

Cockayne, E. J. and Mynhardt, C. M. "The Sequence of Upper and Lower Domination, Independence and Irredundance Numbers of a Graph." Disc. Math. 122, 89-102, 1993).

West, D. B. Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2000.


4.4: Path Independence - Mathematics

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Math Insight

Our goal is to determine if the vector field egin dlvf(x,y) = left( frac<-y>, frac ight) end is conservative (also called path-independent).

One condition for path independence is the following. For a simply connected domain, a continuously differentiable vector field $dlvf$ is path-independent if and only if its curl is zero.

Since $dlvf(x,y)$ is two dimensional, we need to check the scalar curl egin pdiff - pdiff. end We calculate egin pdiff &= frac<1> - frac <(x^2+y^2)^2>= frac<(x^2+y^2)^2>\ pdiff &= -frac<1> +frac <(x^2+y^2)^2>= frac<(x^2+y^2)^2>. end Since these partial derivatives are equal, the curl is zero.

Can we conclude $dlvf$ is conservative? The problem is that $dlvf$ is not defined at the origin $(0,0)$. Its domain of definition has a hole in it, which for two-dimensional regions, is enough to prevent it from being simply connected. The test does not apply, and we still don't know whether or not $dlvf$ is conservative.

Let's try another test, this time a test for path-dependence. If we can find a closed curve along which the integral of $dlvf$ is nonzero, then we can conclude that $dlvf$ is path-dependent. If the curve does not go around the origin, then we can use Green's theorem to show the integral of $dlvf$ zero, egin lint= int_dlr left(pdiff-pdiff ight) dA = int_dlr 0 dA = 0, end as the vector field is defined everywhere in the region $dlr$ inside the closed curve $adlc$.

We must try a closed curve where Green's theorem does not apply, i.e., one that goes around the point at the origin where $dlvf$ is not defined. We'll use the unit circle.

A counterclockwise parametrization of the unit circle is $dllp(t)=(cos t, sin t)$ for le t le 2pi$. On the unit circle, $dlvf$ takes a simple form, egin dlvf(dllp(t)) &= dlvf(cos t, sin t) = left(frac<-sin t>, frac ight) &= (-sin t, cos t). end Therefore, egin dlint &= dplint &= int_0^ <2pi>(-sin t, cos t) cdot (-sin t, cos t) dt &= int_0^ <2pi>(sin^2 t + cos^2 t) dt = int_0^ <2pi>1, dt = 2pi. end

The hole in the domain at the origin did end up causing trouble. We found a curve $dlc$ where the circulation around $dlc$ is not zero. The vector field $dlvf$ is path-dependent.

This vector field is the two-dimensional analogue of one we used to illustrate the subtleties of curl, as it had curl-free macroscopic circulation. The circulation can be clearly seen by plotting the vector field $dlvf$. It's difficult to plot, because the vector field blows up at the origin.

Further insight into $dlvf$ can be obtained from the fact that $dlvf$ has a potential function if, for example, you restrict yourself to the right half-plane with $x>0$. In this case, you can write $dlvf(x,y) = abla f(x,y)$, where $f(x,y) = arctan(y/x)$. Of course, this potential function cannot be extended to the whole plane, or we'd run into a contradiction with the fact that $dlint e 0$ when $dlc$ is the unit circle. Besides not being defined along the line $x=0$, it is also discontinuous across that line. But, the existence of this potential function explains why the curl should be zero away from the line $x=0$. (The whole line $x=0$ isn't special for $dlvf$, as the origin is the only point that causes problems. You could, for example, use $f(x,y) = -arctan(x/y)$ for a potential function away from the line $y=0$.)


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4.4: Path Independence - Mathematics


Professor Email: [email protected]
Department of Mathematics Tel. (530)754-0493 (O)
University of California at Davis Fax: (530)752-6635

  • Research Interests:
    • Geometric measure theory and its applications
    • Optimal transportation and its applications in e.g. mathematical biology, mathematical economics.
    • Publications

    On landscape functions associated with transport paths. Discrete and Continuous Dynamical Systems - Series A, Vol. 34, No. 4, (2014) .

    On the ramified optimal allocation problem. (With Shaofeng Xu). arXiv:1103.0571v1, Networks and Heterogeneous Media p591 - 624, Volume 8, Issue 2, June 2013 .

    On the transport dimension of measures. (With Anna Vershynina ). SIAM J. MATH. ANAL. Vol. 41, No. 6,(2010) pp. 2407-2430.

    The formation of a tree leaf. ESAIM Control Optim. Calc. Var. 13 (2007), no. 2, 359--377.

    Interior regularity of optimal transport paths. Calculus of Variations and Partial Differential Equations . Vol. 20, No. 3 (2004) 283-299.

    Intersection homology theory via rectifiable currents. Calculus of Variations and Partial Differential Equations. Vol. 19, No. 4 (2004), 421-443.

    Conformal deformation of a closed Riemannian submanifold to a minimal submanifold. (with Xu, Senlin) Journal of Mathematical Study , Vol 31 (1998), no. 2, 109--115. A summary version is also published on Chinese Science Bulletin, Vol 43 (1998), no. 6, 527.

    On the spectrum of Clifford hypersurface. (with Xu, Senlin) Journal of Mathematical Study . Vol 29 (1996), no. 4, 5--9.


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