Skip to main content
Mathematics LibreTexts

4.4: Path Independence

  • Page ID
    6483
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    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.

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

    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,

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

    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.

    imageedit_4_7910359843.png


    This page titled 4.4: Path Independence is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.