BSc Mathematics SEM VI 2016 17 2016-17 Analysis & Multivariable Calculus II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 All questions are compulsory
2. Q2 Figures to the right indicate marks 1 (a) Attempt any ONE of the following (8) = L and if f(x,y) Both exists, then prove that d Give an example to show that the converses
   • (ii) Let S be an open subset and f,g : S
   • (b) Attempt any TWO of the following
   • (i) Let +: be defined by. = —6 definition show that each component of f 2)
   • (i) be defined by = then. ‘show and f,(0,0) do not exist. Check whether fis continuous be
   • (iv) Using definition, discuss of f
3. Q2 (a) Attempt any ONE of the following LS Let S be subset. of IR’ and f:S > be that Dif, Dof exists on S. If are continuous on then:show that (a,b) = 8 marks
   • (ii) Let S be an open subset of R” and f:5 R be differentiable at a with total derivative Df (a). Show that f for all y IR" and
   • (b) Attempt any TWO of the following 12
   • (i) and Mean Value Theorem for a scalar field of f at A in the : and:in the direction of AC is 26. Find the directional derivative of < in the direction of AD
   • (q)Verify that = where. u = y ,v =2xsinxsiny . Use chain rule ONE: of the following (8)
   • (i). State prove Theorem for an oriented smooth simple parametrized surface in bounded by simple ,closed, curve traversed counter clockwise assuming general form of Theorem State Divergence Theorem for a solid in 3- space bounded by an closed surface with positive orientation and prove the Divergence theorem for cubical region

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