BSc Mathematics SEM V 2019 20 Oct 2019-20 MATHS 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 the correct alternative in each of the following: An expression for f(x, y)dydx in which the order of integration is 20 marks
   • (c) asum of two integrals (d) None of these li. volume of the solid given by x7 + y? < 1 and <z< 27 1s
   • (c) 1 (d) _ None of these lil, f(x,y) = k, k constant and R = x [c, d] then Sf, equals
   • (c) (d) data insufficient line integral F = and C:x* +y?
   • (a) depends on a
   • (b) does not exists as Green’s Theorem is not applicable
   • (c) is aconstant independent of a
   • (d) none of these
   • v. The image of [0,1] under the transformation f:IR > R? which is defined as
   • (a) arc of a circle (b) An arc of a parabola
   • (c) arc of a hyperbola None of these
   • vi. ydx + xdy along every closed curve C is
   • (c) (d) None of these
   • vii. Let F = where P,Q,R are continuously differentiable and S is the surface given by z= g(x,y),(x%,y) then Sf, F. is given by
   • (c) ag ag (d) None of these
   • viii. The surface integral of F(x,y) = —yi +xjf on S where S is the disc in the XY plane with radius 2 oriented upwards and at the origin is
   • a) 1. b)-1. c) 0. d) None of these
2. Q9 surface integral dS over a closed surface with volume V is
   • a) V b) 3V d) None of these
   • x. A vector field F is tangent to the boundary of a region S in space. Then
   • (a) Gauss Theorem is (b) 0
   • (c) depends only on S (d) None of these
3. Q2 a) Attempt any ONE question from the following: 8 marks
   • i. Define the double integral of a bounded function f where S = [a, b] x [c, d] is a rectangle in R? Further show with usual notations
   • ii. that a continuous function is integrable for a rectangular domain in R?
   • b) Attempt any TWO questions from the following: 12
4. Q1 State the change of variable formula for triple integral clearly stating the conditions under which it is valid. Express further, how will you use to express the triple integral in spherical coordinates the integral J **" dydx by reversing the order of Using a suitable change of variable, evaluate ff, (x* + y*) dxdy where S is the region in the XY-plane bounded by the curves
   • iv. Find the mass for a plate bounded by x = 0,x = 1,y = 0,y = 2 whose density is = x. [hint: mass = Attempt any ONE question from the following: (08)
   • i. If is continuously differentiable function defined on a simply connected region D in IR*, then show that Pdx + Qdy = 0 around every closed curve C in D if and only if =
   • ii. and prove Green’s theorem for a rectangular region
   • b) Attempt any TWO questions from the following: 12
   • i. If f is continuously differentiable scalar field defined on an open set U in R” and C is smooth, closed curve in U with parameterization r(t),t [a,b], then prove that = 0 Using Green’s Theorem, find the area of the region D which is. bounded by lines y = 1, y = 3,x = 0 and the parabola y? = x ili. Verify Green’s Theorem for the function F(x, y) = (2xy + over the region D bounded by positively oriented curve C formed by parabolas
   • iv. Find whether the force field F(x,y,z) = (ysinz,xsinz,xycosz) is conservative. If so find @ so that F = V@ and calculate the work done in the moving the particle form the point P(0,0,0) to the point Q(7,7, 7)
5. Q4 a) Attempt any ONE question from the following: 8 marks
   • i. State and prove the Stokes’ Theorem for oriented smooth, simple parameterized surface in R? bounded by a simple, closed curve traversed counter clockwise assuming general form of Green’s Theorem Let S =r(T) be a smooth parametric surface described by a differentiable function r defined on region T. Let f be a scalar field defined and bounded on S. Define surface integral of f over S. If R and r are smoothly equivalent functions, R(s,t) (G(s,t)) where G(s,t) = u(s,t)i+ v(s,t)j being continuously differentiable. Then show that R, x R; = X Ty) Further prove that fds = Dee) fdS where G(B) =A
   • b) any TWO questions from the following: 12
   • i. Find. the surface area of S where S is the part of the plane x +2y +z that lies inside the cylinder x? + = 4
   • ii. Evaluate the surface integral of vector field F over S where = (x,y,z) and is the paraboloid z = x* + y* —1,-1 <z< 0 oriented Prove the following identities, assuming S and V satisfy the conditions of the Divergence Theorem and scalar fields f and g, components of F have continuous second order partial derivatives, is unit outward normal to S Evaluate the surface integral of the vector field F(x, y, z) = x) over the unit sphere x* + + = 1 using the Gauss divergence theorem Attempt any FOUR questions from the following: (20) ) s where region S is bounded by the three co-ordinate planes and
   • b) area of the region S which is bounded by the parabola x = 9 — y? = 0
   • c) the line integral of the vector field F (x, y,z) = (xz,y + z,x) along the
   • d) the integral of the scalar field f(x,y) y along the line segment from (0,0) to 270)
   • e) Evaluate surface integral of a scalar field f over S where S is the part of the plane x + 2y + 3z = 6 in the first octant
   • f) Using Stokes’ Theorem evaluate the line integral where = (Ay, 2z, 6y) and C is the curve of intersection of

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