Independence of Path

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Section 16.4 Independence of Path

RECALL: \(\qquad\displaystyle\Int_a^b{df\over dx}\,dx = f \Big|_a^b\) .

Figure 16.4.1. Two different paths between the same two points.

The assertion above is the Fundamental Theorem of Calculus, which just says that the integral of a derivative is the function you started with. We could also write this statement simply as

\begin{equation*} \int df = f . \end{equation*}

But recall the master formula (10.1.2), which says

\begin{equation*} df = \grad{f} \cdot d\rr . \end{equation*}

Putting this all together, we get the fundamental theorem for line integrals, which says that

\begin{equation} \Lint \grad{f} \cdot d\rr = f \Big|_A^B\tag{16.4.1} \end{equation}

for any curve \(C\) starting at some point \(A\) and ending at some point \(B\text{.}\) This situation is illustrated in Figure 16.4.1.

Notice that the right-hand side of (16.4.1) does not depend on the curve \(C\text{!}\) ¹ This behavior leads us directly to the notion of path independence. A line integral of the form \(\Lint\FF\cdot d\rr\) is said to be independent of path if its value depends only on the endpoints \(A\) and \(B\) of the curve \(C\text{,}\) not on the particular curve connecting them. If a line integral is independent of path, we no longer need to specify the path \(C\text{,}\) so we instead write

\begin{equation*} \Int_A^B \FF \cdot d\rr . \end{equation*}

If you know that a line integral is independent of path, you may choose a different path (with the same endpoints) which makes evaluating the integral as simple as possible!

There is some fine print here: the curve must lie in a connected region (one with no holes), on which \(\grad f\) is defined everywhere. This condition poses no problem if \(f\) is differentiable everywhere, which it often is, but there are important examples where this assumption fails.