Question ( marks) Each of the six diagrams in Figure represents the graph of some function of two variables whose second partial derivatives are continuous on R. Diagrams show the graphs in a neighbourhood of (, ) together with respective tangent planes at (, ). For each of the functions, one computes the determinant of the Hessian matrix at the origin, that is, f xx (, ) f yy (, ) f xy(, ). It turns out that this expression is positive for two of the six functions, is zero for two of them, and is negative for two of them. Determine which diagrams represent positive Hessian, which ones represent negative Hessian, and which ones represent zero Hessian. 3 3 5 8.5.5.5.5.5.5.5.5 8.5.5.5.5.5.5.5.5 (A) (B) (C) 8.8.9..8...7...5.5.5.5.5.5.5.5.5...8.5.5.5.5 (D) (E) (F) Figure : Graphs of functions from Question. Answer Positive: C, D; zero: B, E; negative: A, F. :)
Question (5 marks) Let f (x, y) = xy x + y. (i) Prove that the limit of f (x, y) as (x, y) approaches (, ) along any straight line is. (ii) Prove that, nevertheless, lim (x,y) (,) f (x, y) is not defined. Solution (i) We need to check lines y = kx and x =. At y = kx we have f (x, kx) = k x 3 = k x, so the limit as x is. Further, f (, y) =, x +k x +k x so the limit as y is. (ii) Let s find the limit along the parabola x = y. We have f (y, y) = y =, y +y so the limit as y is. Since limits along lines x = and x = y are different, the limit of the whole expression is not defined. :)
Question 3 (5 marks) Assume that a function f : R m R satisfies the following conditions: (a) all its second partial derivatives are continuous on R m ; (b) f is harmonic, that is, f x x + f x x + + f xm x m = on R m ; (c) f is Morse, that is, all its critical points are non-degenerate. Prove that f doesn t have a local maximum or a local minimum. Hint: recall how the trace of an m m matrix is determined by its eigenvalues. Solution Given a critical point of the function f, let λ,..., λ m be eigenvalues of the Hessian matrix H f at this point. We have, λ + + λ m = f x x + + f xm x m =. On the other hand, at a local maximum we would have λ,..., λ m < and λ + + λ m <, so it cannot happen. In the same manner, at a local minimum we would have λ,..., λ m > and λ + + λ m >, so it cannot happen either. :) 3
Question ( marks) Evaluate the integral V dxdydz + (x + y ), where V is given by inequalities y and z x + y. Solution In cylindrical coordinates, the integral is π dθ r dr rdz + r = π 8 r 3 dr + r = π [ arctan(r ) ] 8 = π arctan :)
Question 5 ( marks) Express the area bounded by the curve x 5 + y 7 = as an integral in variables p, α, where x = p 5 cos 5 α, y = p 7 sin 7 α. You are not required to evaluate the integral, just write it down. Solution First, notice that the given substitution is the composition of polar coordinates and taking the 5th power of x and the 7th power of y. Hence the given area is θ π, p and the substitution is one-to-one inside it. Thus we need to integrate the absolute value of the Jacobian, which is 5p cos 5 α 5p 5 cos α sin α 7p sin 7 α 7p 7 sin α cos α = 35p cos α sin α, over the given region. The answer is 35 π dα p cos α sin αdp. :) 5
Question ( marks) Find the work of the vector field ( e x + y ) i + ( ln( + y 7 ) + 3z ) j + ( cos(9πz) + 5x ) k over the circle x + y = 9, z = oriented counterclockwise if viewed from top. There are two solutions because the question has an error noticed only during the actual exam. The first solution is the correct one (due to the error) and the second solution is the intended one. Both of them will be considered correct because the error is not obvious and was missed by a number of checks. Solution The answer is not defined because the vector field does not exist for y. :) Solution We ll apply the Stokes Theorem. First, the curl of the given vector field is i j k x y z = 3i 5j k. e x + y ln( + y 7 ) + 3z cos(9πz) + 5x By Stokes Theorem, the work equals the flux of the curl across the disk x + y 9, z =. It is parameterized by x, y with z =, so the normal vector is k and the flux is ( 3i 5j k) kdxdy = dxdy, x +y 9 x +y 9 which is times the area of the disk. Thus the answer is 8π. :)
Question 7 ( marks) Find the flux of the vector field (e y z)i + (sin x + z)j + x 5 yk across the torus given by x = (b + a cos ψ) cos ϕ, y = (b + a cos ψ) sin ϕ, z = a sin ψ, where ϕ, ψ π are parameters and < a < b are some constants (see Figure ). Figure : Torus with a =, b = 5. Answer Let F = (e y z)i + (sin x + z)j + x 5 yk, let T be the given torus, and let V be the solid torus enclosed by T. Then By the Gauss Theorem, div F = (e y z) x + (sin x + z) y + (x 5 y) z =. T F dt = V div Fdxdydz =. :) 7