Multivariate Calculus — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

(b) Show that the vector field F(xy) = (*+2y)i+(2x+y)) is conservative and find a scalar potential f for F. Evaluate the line integral J. FaR, where C is any smooth path connecting A(0,0) to B(1,1). (c) Use Green's theorem to find the work done by the force field When an object moves once counterclockwise around the ellipse 4x*+y2=4. 6. (a). Evaluate Jl(*'+y?)dS, where s is the surface of the hemisphere z=v1-x'- y? (b) Let S be the portion the plane xty+z = 1 that lies in the first octant, and let C be the boundary of S, traversed counterclockwise. Verify Stokes' theorem for the surface S and the vector field F= . (C) Use the divergence theorem to evaluate. Where, F = xyzi + xyzj + xyzk, S is the surface of : the box 0≤x≤1,0≤y ≤2,0≤z≤3, with outward. unit normal vector N. 1. (a) Consider the function: Find the value of A, which will make fcontinuous at the (0,0). (b) The temperature at a point (*, y) on a given metal plate in the xy plane is determined according to the formula T (x,y) = x*+2xy +y degrees. Compute (c) A juice can is 12 cm tall and has a radius of 3 cm. A manufacturer is planning to reduce the height of the can by 0.2 cm and the radius by 0.3 cm. Use a total differential to estimate the percentage decrease in volume that occurs when the new cans are introduced. (Round to the nearest percent.) 2. (a) Letw = f(t) be a differentiable function oft, where t = (x2+y2+z2)1/2. show that (b) (i) Compute double integral by converting to polar coordinates: (ii) Use a double integral to find 'the area bounded by r = cos 2 © , (c) (i) Evaluate Triple integral: D where, D is bounded by the paraboloid z = x2+y2 and the plane z = 1. (ii) Compute the iterated triple integral. (a) Find the volume of the sofid bounded by cylinders y = z2 and y = 2 - z2 and the planes x = 1 and x = -2 (b) Evaluate triple integral dxdydz (b) Find the directional derivative of f (x,y)` = ex2y2 at Po(.1, -1) in the direction toward Q (2,3). (c) 4. Find the absolute extrema of the function f (x,y) = ry-2x-5y on .the closed bounded set S, where S is the triangular region with vertices (0,0) {7,0) 8. (a) (i) Find the volume of the solid bounded below by the rectangle R in the xy-plane and above by the graph: where, D is the solid sphere xa+y2+z2 < 3 Use suitable change of `variable find the area of the region R bounded by the hyperbolas zry = 1 and xy = 4 and the lines y = x and y = 4x. (ii) Sketch the region of integration and compute the double integral: 5. (a) Evaluate JCF dR, where F = ly2-z8)i+(2yz)j-x2k and C is the curve defined parametrically by x = t2,y = 2t and z = t, for 0 < ts 1. fo:toy2:sin?dxdy