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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Activity 13.4.5 .
Suppose that
\(\vF\) is a continuous path-independent vector field (in
\(\R^2\) or
\(\R^3\) ) on some region
\(D\text{.}\)
(a) Let
\(P\) and
\(Q\) be points in
\(D\) and let
\(C_1\) and
\(C_2\) be oriented curves from
\(P\) to
\(Q\text{.}\) What can you say about
\(\int_{C_1}\vF\cdot d\vr\) and
\(\int_{C_2}\vF\cdot d\vr\text{?}\)
(b) Let
\(C = C_1 - C_2\text{.}\) Explain why
\(C\) is a closed curve.
(c) Calculate
\(\oint_C\vF\cdot d\vr\text{.}\)
(d) Write a sentence that summarizes what we can conclude about line integrals of
\(\vF\) at this point in the activity.
(e) Now let us suppose that
\(\vG\) is a continuous vector field on a region
\(D\) for which
\(\oint_C\vG\cdot d\vr = 0\) for all closed curves
\(C\text{.}\) Pick two points
\(P\) and
\(Q\) in
\(D\text{.}\) Let
\(C_1\) and
\(C_2\) be oriented curves from
\(P\) to
\(Q\text{.}\) What type of curve is
\(C = C_1 - C_2\text{?}\)
(f) What is
\(\oint_C\vG\cdot d\vr\text{?}\) Why?
(g) What does that tell you about the relationship between
\(\int_{C_1}\vG\cdot d\vr\) and
\(\int_{C_2}\vG\cdot d\vr\text{?}\)
(h) Explain why this shows that
\(\vG\) is path-independent.
