LSweek31

LSweek31 - MATH1211/LSweek31/TNK/10-11(2) THE UNIVERSITY OF...

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1 MATH1211/LSweek31/TNK/10-11(2) THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211: Multivariable Calculus Lecture Summary – Week 31 March 28, 2011 Section 6.1 . We recalled Definition 1.1 (p.364) of the scalar line integral. We noted that is the notation for the scalar line integral, but in order to compute it we need to write it as so that we can integrate w.r.t. , where is the parameter of the path . We noted that the path of the line integral can be piecewise , which is a union of a finite number of paths joining together (p.365). We learned Definition 1.2 ( vector line integral ) (p.366). We noted the difference between its notation and that of the scalar line integral: for the scalar line integral, the thing to be integrated is where is a scalar valued function and is the arc-length element along the path; for the vector line integral the thing to be integrated is the dot product , where is a vector valued function, and is a vector element along the path. We noted that the vector line integral has at least three different formulations: , where everything is written in terms of the parameter of the path . , where denotes the unit tangent vector along the path (p.367). , where + . (More generally, if is a vector field on where each is a scalar function, then see p.368). We went through Example 5 (p.369) We learned reparametrization of paths ( Definition 1.3 , p.369), and that the scalar line integral is not changed when the path is reparametrized ( Theorem 1.4 , p.370). The same is true for vector line integrals if the orientation of the path keeps the same as before, but the value of the vector line integral will become negative of the original one if the orientation of the path is reversed (
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LSweek31 - MATH1211/LSweek31/TNK/10-11(2) THE UNIVERSITY OF...

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