BSc Mathematics SEM V 2018 19 2018-19 Maths 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: 20 marks
   • i. If D is the unit sphere x? + y* + z* < 1 then is equal to
   • (c) (d) None of these
   • ii. The volume V of the solid above the region R= {(r, and under the surface z = e
   • (a) me (b) il. f(x,y) = k, k constant and R = x [c,d] then SS kdA equals
   • (c) (d) data insufficient
   • iv. A parameterization a of a circle of radius 2 centered at the origin in the X Z plane is given by J, 2ydx + 2xdy where C is the path (t?, < 1 Then
   • (c) (d) None of these P dx + Q dy = 0 around every closed path C in a simply connected region
   • (a) By if P and Q are function
   • (b) as always
   • (c) By
   • (d) None of the above
   • vii. surface area of the triangle with vertices (1,0,0), 1) is
   • viii. The fundamental vector product for the cone
   • (c) r) (d) None of these
   • (c) 1&3 (d) None of these
   • x. The surface integral axi + byj + over the surface of a unit sphere enclosing a volume V is
   • (c) (a+b+c)4n? (d) None of these
2. Q2 a) Attempt any ONE. State and prove Fubini’s Theorem for a rectangular domain in 8 marks
   • ii. If U isanopensetin R* containing the rectangle [a,b] x [c,d] and f:U > R is continuously differentiable function then show that
   • b) any TWO. < y is a region in R? where 12
   • b] R are continuous and a function f: S > R is continuous in the interior of S with f (x,y) = 0 then prove that ff. 5 f 29 the integral (9 — 2dydx by reversing the order of lil. the integral where S is the solid in the first octant bounded by the sphere x? + y? + = 9
   • iv. Using cylindrical co-ordinates find the volume of the solid region S in R? bounded by the cone z = y? and the paraboloid z = x* + Attempt any ONE. (08) Suppose F is a continuous vector field defined on an open connected set U in R”. Define a function > R by = F where is a fixed point in is conservative. Show that V@(v) = F(v) Vv U
   • ii. State and prove Green’s Theorem for a rectangle. Further state Green’s theorem for a closed region D in R? whose boundary is a simple closed curve C. Show that area of region D = xdy
   • b) Attempt any TWO. Evaluate the line integral + 2xy)dx + (x? + 2xy) dy ij, Using Green’s theorem evaluate the line integral whereC is the positively oriented boundary of the region R enclosed between y = x? andy =x ili. Define the line integral of a vector field F defined on an open set U in R” along an oriented curve in U. If and equivalent but orientation reversing curves in U, show that F 12
   • iv. Find the work done by the force F = (—4xy, 8y,z) as the point of application moves along the curve of intersection of the parabolic cylinder y = x* and the plane z = 1 from to (2,4,1)
3. Q4 a) Attempt any ONE. 8 marks
   • i. Let S =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 rare smoothly equivalent functions, continuously differentiable. Then show (4) Se (B) fds li. State and prove Stokes’. Theorem for an oriented smooth, simple parameterized surface in R? bounded by a simple, closed curve traversed counter clockwise assuming general form of Green’s Theorem
   • b) Attempt any TWO. IfS and C satisfy hypothesis of Stokes’ Theorem and f, g have continuous second order partial derivative, prove with usual notations 12
   • ii. Evaluate surface integral of f(x, y,z) = x* + y* where S is the surface of the paraboloid x* + y* = 4 —z above the XY-plane iit. Use Stokes’ theorem to find (curl F) -n dS where = and S is the surface of the paraboloid z = 1 — x* — 0
4. Q4 = Evaluate ff, f(x,y,z).fids where f(x,y,z) and S is the surface of the cylinder x* + y? = 4 between 0<z <4 Attempt any FOUR. (20) aV where region S is bounded by the three co-ordinate planes and Evaluate by converting into polar coordinates
   • c) Evaluate the line integral of f(x,y,z)=x+y+z, along the path
   • d) Find a potential function of F where
   • e) Find surface area of S where S is the surface of the sphere x? + + = 16 in
   • f) Use Gauss Divergence theorem to find ff. s F : where — x) and S is the cube bounded by the planes

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