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HARVARD MATH 21A - SCALAR LINE INTEGRALS

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4/7/2004, SCALAR LINE INTEGRALS Math21a,O. KnillSCALAR LINE INTEGRALS.If f(r(t)) is a function defined on a curve γ : t 7→ ~r(t), thenRbaf(~r(t))|~r0(t)| dtis called the scalar lineintegral of f along the curve γ.NOTATION. The short-hand notationRγf ds is also used.WRITTEN OUT. If f(x, y) is the function and ~r(t) = (x(t), y(t)), we can writeRbaf(x(t), y(t))px0(t)2+ y0(t)2dt.In three dimensions, where ~r(t) = (x(t), y(t), z(t)), we can writeRbaf(x(t), y(t), z(t))px0(t)2+ y0(t)2+ z0(t)2dt.EXAMPLE. Integrate f(x, y, z) = x2+ y2+ z2over the path r(t) = (cos(t), sin(t), t) from t = 0 to t = π. Theanswer isRπ0(1 + t2)√2 dt = π√2 + π3√2/3.EXAMPLE: Let r(t) = {cos(t), sin(t), t2/2} be the path of a model plane. What is the average height of theplane?This is not a very clearly formulated question. We want to knowZ2π0z(t)|r0(t)| dt =Z2π0t22p1 + t2dtif we want to know the average height of the path andZ2π0z(t) dt =Z2π0t2/2 dtif we want to know the average height per time.EXAMPLE. A wire r(t) = (cos(t), 0, sin(t)) has thickness f (r(t)) =sin2(t) and t ∈ [0, 2π] What is the mass of this wire? The mass is,because r0(t) = 1:M =Z2π0sin2(t) dt = π .EXAMPLE. One of the hits on the web in March 2004 was a photo report of a Russian girl ”Elena” whorode with her Kawasaki motorcycle and a camera through the deserted Chernobyl area and left an impressivedocument on the web. The URL is http://www.angelfire.com/extreme4/kiddofspeed/Assume Elena picks up radioactive radiationproportional to the radioactivity level f (x, y)and the amount of path covered with the bike,then the total radiation obtained during thetime [0, T ] isZT0f(r(t))|r0(t)| dt .This is not realistic. If Elena stops, she wouldget no radiation increase. The correct integralwould rather beZT0f(r(t)) dt .REMARK.Scalar line integrals should be thought as a generalization of the length integral.Do not mix it up with the line integral defined by a vector field which we cover next and which is infinitelymore important.The examples above show that dealing with scalar line integrals can be confusing. You are measuring quantitieswith it which are given ”per distance” and not quantities ”per time”. Scalar integrals hardly appear inapplications. (They do for example in tomography where the problem is to reconctruct f (x, y, z) from knowingall line integrals along all lines. But also there, it is possible and simpler to avoid them.)• the application of computing mass is very artificial because mass is a triple integral. All solid bodies, evenwires have three dimension.• In general, for one dimensional situations, the density is constant so that the scalar line integral is actuallya usual length integral. Nobody questions the importance of the length integral.• For ”center of mass” or moment of inertia” computations, one better uses triple integrals. Arcs or wireshave a nonzero radius and the ”simplification” done by computing it with one dimensional integralsproduces an error. For mass computations, the error is zero by the Pappus Centroid theorem.Introducing artificial scalar line integrals is not only confusing, it is also not precise.The topic has been introduced into calculus text books as a ”bridge” to ease the transition from 1D integralsto line integrals but experience shows that this ”bridge” unnecessarily complicates things because it introducesa new concept.Just treat the topic as a ”footnote” to the length


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HARVARD MATH 21A - SCALAR LINE INTEGRALS

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