prf - 1 Mathematics 23B Fall 2008 V Ginzburg Practice Final...

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1 Mathematics 23B; Fall 2008; V. Ginzburg Practice Final 1. For each of the ten questions below, state whether the assertion is true or false . (You do not need to justify your answer.) (a) The area of the portion of the graph of z = f ( x, y ) over a region D in the ( xy )-plane is equal to ZZ D s 1 + ∂f ∂x 2 + ∂f ∂y 2 dA. (b) Let Ω be the region in R 3 bounded by a surface S and F a C 1 vector field in R 3 . Gauss’ theorem asserts that ZZZ Ω ∇ · F dV = ZZ S F · d S , where S is oriented inward. (c) Let S be the unit sphere. Then RR S x 2 z dS = 0. (d) Let z , r , and θ be cylindrical coordinates. Then ( x, y, z ) ( r, θ, z ) = r 2 sin θ. (e) Every onto map is necessarily one-to-one. (f) The integral RR S f dS , where f is a function, changes sign when the orien- tation of S is changed. (g) Let D be the region to which Green’s theorem applies. Then the area of D is equal to 1 2 R ∂D ( xdy - ydx ). (h) Let a surface S be oriented by the unit normal n . Then RR S F · d S = RR S ( F · n ) dS , for any vector field F .
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