Nagoya University, G30 program                                                            Spring 2021
           Calculus II: Final examination   b                                          Instructor : Serge Richard
           Exercise 1 Reply Yes or No to the following questions:
              i) Are you going to cheat ?
             ii) If you cheat, are you aware of the legal consequences ? and that you won’t be trustable anymore ?
           Exercise 2 Consider the map
                                       f : R2 ∋ (x,y) 7→ sin(x + y) − cos(xy) + 1 ∈ R.                        (1)
             (i) Show that the implicit function theorem can be applied at the point (0,0) ∈ R2,
             (ii) Determine the equation of the tangent at the point (0,0) of the curve of equation f(x,y) = 0.
           Exercise 3 Compute the curve integrals (all steps of your computations are necessary) for :
                      2                  2              2                                            2
             (i) f : R ∋ (x,y) 7→ (2xy,x +2y +3) ∈ R and the curve defined by the parabola y = x from (0,0)
                 to (1,1),
             (ii) f : R3 ∋ (x,y,z) 7→ (x,z,x − y) ∈ R3 and the curve defined by the segment between (0,0,0) and
                 (1,2,3).
           Exercise 4 Compute the following integrals (all steps of your computations are necessary) :
                 RR                              √   √ 
              i)   Ω xcos(xy)dxdy with Ω = 0, π × 0, π ,
             ii) RR   ydxdy with Ω the triangle defined by the lines y = 0, x = 0, y = x+1,
                   Ω
                 RRR    2    2
             iii)   Ω(x +y )zdxdydz with Ω the unit upper half-ball centered at 0.
           Exercise 5 Check the validity of the divergence theorem in R3 by computing separately both sides of
           the equality of this theorem:
              i) For a unit ball centered at 0 and for the vector field f : R3 → R3 defined by f(x,y,z) = (x,y,z),
             ii) For a vertical cylinder (together with its upper and lower faces) defined by x = cos(θ), y = sin(θ)
                                                             3      3                                

                 for θ ∈ [0,2π] and z ∈ [0,1], and for f : R → R defined by f(x,y,z) = 0,0,g(z) for some
                 g : R → R of class C1.
           Exercise 6 Consider the function defined by (1) of Exercise 2.
              i) Show that there exist 2 families of critical points for this function,
             ii) Show that (π/2,0) is a critical point and discuss if it is a local extremum or a saddle point,
             iii) Show that (0,−π/2) is a critical point and discuss if it is a local extremum or a saddle point.
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