LSweek33

LSweek33 - MATH1211/LSweek33/TNK/10-11(2) THE UNIVERSITY OF...

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1 MATH1211/LSweek33/TNK/10-11(2) THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211: Multivariable Calculus Lecture Summary – Week 33 April 11, 2011 We redrew Figure 7.13 in Section 7.1 (p.414) as shown below. The image of the small rectangular region in D with sides given by vectors ¢ s i and ¢ t j under the mapping X is approximately equal to the parallelogram with sides ¢ s T s and ¢ t T t , the area of which is given by k ¢ s T s kk ¢ t T t k sin μ = ¡ k T s kk T t k sin μ ¢ ¢ s ¢ t = k T s £ T t k ¢ s ¢ t; where μ is the angle made by the vectors ¢ s T s and ¢ t T t . Thus the total surface area of S = X ( D ) is approximately equal to P i;j k T s ( s i ¡ 1 ;t j ¡ 1 ) £ T t ( s i ¡ 1 ;t j ¡ 1 ) k ¢ s i ¢ t j where the sum is taken over all small rectangles resulted from partitioning D . By taking limit of this sum when all ¢ s i ; ¢ t j ! 0 , we obtain that the surface area of S is RR D k T s £ T t k ds dt (see p.414 415). We noted that the surface area formula RR D k T s £ T t k ds dt is analogous to the arclength formula R b a k x 0 ( t ) k dt of a curve parametrized by x ( t ) ;a · t · b . We remarked that in equation (7) on p.416, we may write the term ¡ @ ( x; z ) @ ( s; t ) as @ ( z;x ) @ ( s; t ) . This is based on the fact that, for any square matrix, when two columns (or two rows) are interchanged, the determinant will become negative of the original determinant. Accordingly, we may use the cycle of
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LSweek33 - MATH1211/LSweek33/TNK/10-11(2) THE UNIVERSITY OF...

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