BSc Sem VI 2012 2013 2015 Math 2 Question Paper - Mumbai University | munotes

Questions asked in this paper

• Figures to the right indicate marks for respective subquestions

1. Q2 (a) Let S be an open subset of R? and be such that Dif, exist on If (a,b) S and D2 f are continuous on S, then show that (7)
   • (b) Attempt any two questions:
   • (i) If +R are differentiable on $, where S is an open subset of then show that
   • (2) V = at points where g 0. 4
   • (ii) Find the direction derivatives of = + 2z at (2, 2,1) in the direction of outward normal to the sphere x? + (4)
   • (iii) Let f : R? + R be defined by Show that f is differentiable at (0,0). (4) (Wy) Determine the second order Taylor formula f(z, y) = cos (4) OF differentiable on T, define the fundamental ying on the surface, then show that is
   • (b) Attempt any two questions:
   • (i) the of the Surface of surface of revolution of the curve z = for is given by (4)
   • (ii) Find the equation of the tangent plane to the given parametric surface F(u,v) =
   • (iii) Use Stoke's theorem to calculate the hemisphere = 0 cut by the cone
   • (iv) Assuming S and V satisfy the conditions of the Divergence Theorem, with usual nota- ry tions, prove that (4)
   • (1) = where F = and |V| = volume of V
2. Q4 Attempt any three questions: ‘ Find a constant cso that at any point of intersection of the two spheres 22 — 3 and x? + 1)? 4 22 1, the corresponding tangent planes are perpendicular to each
   • (b) Compute the matrices D(f(9(1,1))) and D(f 9(1,1)) and verify that Let flowy) = tor (0,0). Find lim and Does lim exist? Justify. (5)
   • (d) Verify Stoke’s Theorem for F(z, y, z) = 3yi + — 6rk, S is the part of the paraboloid that lies above the zy-plane oriented upwards. (5)

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