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where G u; v; w  F f u; v; w ; g u; v; w ; h u; v; w . State su cient conditions under which the result is valid. See Problem 10.83. Alternatively, employ the di erential element of volume dV @r @r @r du dv dw (recall the geometric meaning). @u @v @w 10.85. (a) Show that in general the equation r r u; v geometrically represents a surface. (b) Discuss the geometric signi cance of u c1 ; v c2 , where c1 and c2 are constants. (c) Prove that the element of arc length on this surface is given by ds2 E du2 2F du dv G dv2 where E @r @r ; @u @u F @r @r ; @u @v G @r @r : @v @v
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LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS
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10.86. (a) Referring to Problem 10.85, show that the element of surface area is given by dS p EG F 2 du dv. (b) Deduce from (a) that the area of a surface r r u; v is   s        @r @r  @r @r @r @r   @u @v @u @v @u @v
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A B C D A C B D A D B C . 10.87. (a) Prove that r a sin u cos v i a sin u sin v j a cos u, 0 @ u @ ; 0 @ v < 2 represents a sphere of radius a. (b) Use Problem 10.86 to show that the surface area of this sphere is 4a2 . 10.88. Use the result of Problem 10.34 to obtain div A in (a) cylindrical and (b) spherical coordinates. See Page 161.
In nite Series
The early developers of the calculus, including Newton and Leibniz, were well aware of the importance of in nite series. The values of many functions such as sine and cosine were geometrically obtainable only in special cases. In nite series provided a way of developing extensive tables of values for them. This chapter begins with a statement of what is meant by in nite series, then the question of when these sums can be assigned values is addressed. Much information can be obtained by exploring in nite sums of constant terms; however, the eventual objective in analysis is to introduce series that depend on variables. This presents the possibility of representing functions by series. Afterward, the question of how continuity, di erentiability, and integrability play a role can be examined. The question of dividing a line segment into in nitesimal parts has stimulated the imaginations of philosophers for a very long time. In a corruption of a paradox introduce by Zeno of Elea (in the fth century B.C.) a dimensionless frog sits on the end of a one-dimensional log of unit length. The frog jumps halfway, and then halfway and halfway ad in nitum. The question is whether the frog ever reaches the other end. Mathematically, an unending sum, 1 1 1 n 2 4 2 is suggested. Common sense tells us that the sum must approach one even though that value is never attained. We can form sequences of partial sums 1 1 1 1 1 1 S1 ; S2 ; . . . ; Sn n 2 2 4 2 4 2 and then examine the limit. This returns us to 2 and the modern manner of thinking about the in nitesimal. In this chapter consideration of such sums launches us on the road to the theory of in nite series.
DEFINITIONS OF INFINITE SERIES AND THEIR CONVERGENCE AND DIVERGENCE De nition: The sum
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