Topics in Multivariate Calculus — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

1. (a) Define e - 6 definition of limit of a function of two variables. Using this definition, show that * )(0)(73 +y=)3 = 0. (b) (i) Given that the function f(x) = ByE for x' #y2 is continuous at the otherwise origin, what is A? (ii) Let z be defined implicitly as a function of x and y by the equation x=2 + yz3= x. Determine zx and Zy. (c) When two resistances Ry and Rz are connected in parallel, the total resistance R satisfies If Ry is measured as 300 ohms with a maximurn error of 2% and R2 is measured as 500 ohms with a maximum error of 3%, what is the maximum percentage error in R? 2.(a) (i) Define gradient of a function of two variables. (i) In what direction, is the function defined by f(x, y) = xezy- increasing, rapidly at the point Po (2,1), and what is the maximum rate of increase? In what direction is f decreasing most rapidly? (b) (i) Find a vector that is normal to the level surface x? + 2xy - yz + 3z° = 7 at the point P. (1,1, -1). (ii) Let z = f(x, y), where x = at and y = at for constants a and b. Assuming all necessary differentiability, find dz in terms of the partial derivatives of z. (c) Find the point on the plane x + 2y + z = 5 that is closest to the point P(0,3,4). 3. (a) (i) Evaluate R x2+1 2xy dA; R: 0 ≤ x ≤ 1,1 ≤ y ≤ 3 using iterated integration. (ii) Find the volume of the solid bounded below by the rectangle R in the xy plane and above by the graph of=/(xy where/(xy) =2x+3: R:0≤x≤1, 05y = (b) Evaluate JJ, * dy dx ; D is the region between the parabola y = x? and the line y = 2x + 3. (c) Evaluate 53J- ex+y dady by converting to polar coordinates 4. (a) Find the volume of the tetrahedron T bounded by the plane x+y + z = 1 and the coordinate planes x = 0, y = 0, z = 0. (b) Use cylindrical coordinates to evaluate SS, z(x + y2)-1/2 dx dy dz where D is solid bounded above by the plane z = 2 and below by the surface 2z = x? + y? (c) Let D be the region in xy - plane that is bounded by coordinate axes and the line * + y = 1. Use suitable change of variables u = x - y, v = x + y to compute the integral 5. (a) Evaluate the line integral Jc (v dx + x dy + zdz), where C is the helical path given by * = cost, y = sint, z = t for0 ≤ t ≤ n/2. (b) Verify the vector field F(x,y) = (x+ t + (2x / is conservative and find a scalar potential f for F. Then evaluate the line integral J, F.dR, where C is any smooth path connecting A(0,0) to B (1,1). (c) Verify Green's Theorem for the line integral § (2ydx - xdy) where C is the closed path as shown in the figure below 6. (a) Evaluate Sl, (x? + y)dS where S is the surface of the hemisphere (b) Use Stokes' theorem to evaluate J/, (curl F.N)dS, where F = xi + y≥) + xyz k and S is that part of paraboloid z = 4 - x2 - y2 with z ≥ 0. Use the upward unit normal vector. (c) Use the divergence theorem to evaluate the surface integral JJ, F. N dS for F = 2y l-z] + 3x k and S is the surface of the upper five faces of the unit cube 0 ≤ * ≤ 1, 0 ≤