BSc Sem V 2023 2024 Oct 2024 MATHEMATICS MULTIVARIABLE CALCULUS II PAPER SUBJECT CODE Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2) Figures to the right indicate marks

1. Q1 Attempt any one from the following. 8 marks
2. Q1 Define the double integral of a bounded function f : S > R where S = [a, b] x [c, d] is a rectangle in R? Further show with usual notations
   • ii) Let U be an open set in containing the rectangle [a, b] x [c, d] Suppose f: U R is continuously differential function. Show that = ay where g(x) = vx [a,b
   • B) Attempt any two from the following. 12
3. Q1 Prove that a continuous function is integrable for a rectangular domain in
   • ii) Evaluate S = <z
   • iii) (x? + dxdy where S is the region in the XY-plane bounded by the curves x* — y? =1, =2,xy =2,xy any one from the following. (8)
4. Q1 Let U be an open set in R” and @ : [a,b] U be a parameterization of curve I’. If f, g : U > R are continuous function, then prove that where c,d are real constants. Further show that = ft+ f, where I, and I, are restrictions of @ to and wherea<c <b
   • ii) State and prove the Green’s theorem for a rectangle
   • B) any two from the following. 12
5. Q1 Evaluate the line integral of scalar field f(x, y,z) along the curve C with parameterization y(t) = (sint,cost,t), O<t <
   • ii) the Greens’ theorem to evaluate + 3x)dx + (2y — x)dy where C is the ellipse x* + ili) Evaluate the line integral x*dx + xy dy + dz, where C is
6. Q3 Attempt any one from the following. 8 marks
7. Q1 State 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
   • ii) For the surface 7(u, v) described by the vector equation 7(u, v) = differentiable on T, define the fundamental vector product x If C is a smooth curve lying on the surface, C = (t)), > T, then show that x 2 is normal to C at each point
   • B) Attempt any two from the following. 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 = f(x, y) then show that area of S is given by
   • ii) Evaluate surface integral of scalar field f over the surface S where = xyz, S is the surface of the cone z* = x? + between
   • iii) Verify the Gauss Divergence Theorem for = over the surface of the sphere x* + y* + z? = 9
8. Q4 Attempt any three from the following. 15 marks
9. Q1 Reverse the order of integration and evaluate f 0 f 1+ y? dydx
   • ii) State Fubini’s theorem and hence evaluate ff, f dA where f(x,y) = 5xy and D is the region bounded by x = tl andy = +2
   • iii) Evaluate the following integral by showing it is independent on the path
   • iv) Using Green’s Theorem, find the area of the region D where D is the
   • v) Evaluate surface area of the surface S cut from the by the cylinder whose walls are x = y* andx =
   • vi) the Stoke’s Theorem, evaluate the surface integral ff where = (x + S is the surface of the cone + y? above the plane z = 0

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