BSc Mathematics SEM V ATKT May 2018-19 ATKT MATHEMATICS MULTIVARIABLE CALCULUS II Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2. Figures to the right indicate full marks

1. Q1 Choose correct alternative in each of the following: The integral pe f (Vx? + dydx in polar coordinates is 20 marks
   • i. xf (y)dydx where f is continuous function defined on [0,1]. Then J is
   • (a) (y — (b) Independent of f(y) The volume of region bounded by =0,y =0,z =
   • (c) 216 cubic units (d) None of these
   • iv. The parametric equations x = cos(cost),y [0,7] describes
   • (b) of a circle in first quadrant
   • (c) anarc ofa circle in the first and fourth quadrant None of the above The line integral ; F = and C:x* +
   • (a) _ depends ona
   • (b) does not Green’s Theorem is not applicable
   • (c) is a constant independent of ‘a’ None. of the above
   • vi. = 2xyze* i+ ze* j + ye* k and f(0,0,0) = 7. Then f(1,1,2) =?
   • vii. The magnitude of the fundamental vector product x for surface
   • (d) None of these
   • viii. The flux of the vector field 7 = xt + yj + zk across the unit sphere
   • (c) (d) None of these
2. Q9 The surface integral surface S with volume V, where
   • (c) O (d) None of these
3. Q10 div(curl(x?, yz, sin z)) is
   • (c) 0 (d) None of these Attempt any ONE question from the following: (08)
4. Q1 Define the double integral of a bounded function f: S > IR where S = x [c,d] is a rectangular region in R* and using usual notation show that m(b — a)(d—c) < Sf, f
   • ii. and prove Fubini’s Theorem for a rectangular domain in
   • b) Attempt any TWO questions from the following: 12
5. Q1 Prove that every continuous function defined on a rectangular domain D in the integral ff using polar coordinates, where S is the region in the first quadrant bounded above by the circle (x — 1)* =1 and below by the line y = x
   • iii. Using cylindrical co-ordinates find the volume of the solid region S in R? which is bounded by the paraboloid x* + y? = 4 — z and the plane z = 0
6. Q4 Byaluate SS, where is region in R? between the two spheres with centre at the orgin and radii 2 and 5
   • a) Attempt any ONE question from the following: 8
   • i. Let continuously differentiable scalar field defined on an open set U in R”. Suppose are two points of U that can be connected by piecewise smooth curve C lying in U. Prove that = f(Q) — f(P) given that Further if F = where (x,y) = cos(x + 2y) , does there exist a smooth, closed path C such that J. F.dr = 1 ? If so, find such a path C
   • ii. State and prove Green’s Theorem for a rectangle. Evaluate — + (7x + +1) dy where C is the circle
   • b) Attempt any TWO that the vector F = (ysinz, xsin z,xy cosz) is conservative. If so find f such that = Vf 12
   • ii. Using Green’s Theorem, find the area of the region D whose boundary is positively oriented simple closed curve bounded by the lines y = 1,y = 3,x = 0 and the parabola y? = x
   • iii. Show that two equivalent parameterized curves in have essentially the same image set. Show that the converse is not true by considering curves a(t)= (cost, sint) and B(t) =(sint, cost),O<t
   • iv. F = a continuously differentiable function defined on a simply connected region D Show that JS. Pdx + Qdy = every piecewise smooth closed curve C in D if and only if = <2
7. Q4 a) Attempt any ONE. 8 marks
   • i. LetS =7r(T) be a smooth parametric surface described by a differentiable function defined on region T. Let f be defined and bounded on S. Define surface integral of f over S. If R and r are smoothly equivalent functions, R(s,t) =7(G(s,t)) where G(s,t) = u(s,t)i+ v(s,t)j being continuously differentiable. Then show that = (gy S where G(B) = A
   • ii. Divergence Theorem for a solid in 3-space (or R*) bounded by an orientable closed surface with positive orientation and prove the divergence Theorem for cubical region
   • b) Attempt any TWO 12
   • i. Let S =r(T) be a smooth parametric surface in uv plane. Define area of S If S is represented by an equation z = then show that area of S is iven by + dxdy where T is projection of S on
   • ii. Evaluate the surface integral of F(x, y,z) = over S, where S is the hemisphere above XY-plane of radius 2
   • iii. Find surface area of the part of the paraboloid z = x? + y” that lies under
   • iv. Use Stokes’ theorem to evaluate (curl where F(x,y,z) = yi+ xj + xzk where S is the surface of the hemisphere x* + y?4+ z= 0 and nis the unit normal with a non-negative Attempt any FOUR. (20)
   • a) Using double integration, find the area of the region S in R? bounded by the
   • b) Evaluate dzdxdy
   • c) Let U be an open set in R” and a: [a,b] > U be a parameterization of curve I. If f,g : U > Rare continuous functions, then prove that
   • d) Evaluate the integral of the vector field, F(x, y,z) = (x,2— xy, 1) along the circle of radius 1, with centre at the origin and lying in the yz, plane, traversed counterclockwise as viewed from the positive x axis Use Gauss Divergence theorem to evaluate where and S is the region given by
   • f) Evaluate the surface integral of = z over the surface parameterized by

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