BSc Mathematics SEM V 2016 17 2016-17 Maths & Calculus Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 the right indicate marks for the respective parts, function is integrable for a rectangular (8) Is f integrable over R ? Justify your answer Let U be an open set in R2 containing the rectangle x Suppose f 3 — Ris a continuously differentiable that g(x) = ax where g(x) = fe x ela, i State the change of variables formula for triple integral, stating clearly - (12) the conditions under which it is valid. Use it to express S So ” dzdydx in spherical co-ordinates (x? + y?) dxdy where is, the region in the XY-plane bounded by the curves x? — y? = —y* =2,xy=2,xy=4 by using a suitable change of variable Evaluate converting to polar coordinates iv Find the centre of mass of a in the shape of a rectangle ABCD if the density at any point is the product of the distances of the point from two adjacent sides AD
2. Q2 a i isa continuong vector field defined on.an open connected set - U in R”. Define a funvtion > Rby = where Vo is a fixed point in U and F is Show that = Vv EU ii State and prove Green’s Theorem for a rectangle If R is the region enclosed between the circles x? + =1 and x? + = 4, express R dxdy in terms of line integral is setin isa parameterized curve in U, define the (12) line of f along for a continuous function f:U > R. Show furthér, if a, B are two equivalent parameterized curves, then the line integrals of f along them coincide ii Evaluate J. F, where F(x,y,z) = Jyj and C is the segment from to Consider the vector field F(x,y,z) = + Find a scalar function f such that F = Vf. Hence evaluate the line integral ly differentiable on iv are two scalar fields which are continuously differen Find ff, f.g ifu(x,y) = yon the boundary 8 marks
   • T. Let f be defined be? defined of f over R and r are
3. Q3 i tiable function G(s. t)) where G(s,t) differen Define sur th=T Then show that ii State and Pro surface the portion of the scalar the plane z= 1 and z = 2 1 f the vector field F(x, st octant surface integra’ 3y + in the over S which is the surface of of iii Using Stokes’ Theorem of the cylinder — (x2, along C where Cis the = 2y and the plane y = 2. v) = iv For the surface described Z are differentiable on T, define the ‘fundamental vector product is a smooth curve lying on the surface, oc: show that = x = is normal at each
4. Q4 and S is bounded by the parabola and y Sketch the region S of integration. Express Sg fim ( terms of both the integrals
   • d) F=(P, a continuously differentiable function defined on simply region D in R? Show that F is conservative on D if
   • e) S and C satisfy the hypotheses of Stokes’ Theorem and continuous second order partial derivatives. Prove with ‘Find area of S where S is the part of the paraboloid Vv that lies under the plane z = 9

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