-- -*- M2-comint -*- {* hash: 22560454 *} i1 : R = QQ[a,b,t,x,y,z]; i2 : f = x^7 * y^11; i3 : diff(x,f) 6 11 o3 = 7x y o3 : R i4 : diff(y,f) 7 10 o4 = 11x y o4 : R i5 : diff(x^2,f) 5 11 o5 = 42x y o5 : R i6 : diff(x*y,f) 6 10 o6 = 77x y o6 : R i7 : diff(y^2,f) 7 9 o7 = 110x y o7 : R i8 : diff(x+y,f) 7 10 6 11 o8 = 11x y + 7x y o8 : R i9 : diff(x^2+x*y+y^2,f) 7 9 6 10 5 11 o9 = 110x y + 77x y + 42x y o9 : R i10 : m = matrix {{x^3, x^4},{x^5,x^6}} o10 = | x3 x4 | | x5 x6 | 2 2 o10 : Matrix R <--- R i11 : diff(x,m) o11 = | 3x2 4x3 | | 5x4 6x5 | 2 2 o11 : Matrix R <--- R i12 : diff(x^2,m) o12 = | 6x 12x2 | | 20x3 30x4 | 2 2 o12 : Matrix R <--- R i13 : diff(matrix {{x,x^2,x^3,x^4}}, m) o13 = | 3x2 4x3 6x 12x2 6 24x 0 24 | | 5x4 6x5 20x3 30x4 60x2 120x3 120x 360x2 | 2 8 o13 : Matrix R <--- R i14 : diff(matrix {{x,x^2},{x^3,x^4}}, m) o14 = | 3x2 4x3 6x 12x2 | | 5x4 6x5 20x3 30x4 | | 6 24x 0 24 | | 60x2 120x3 120x 360x2 | 4 4 o14 : Matrix R <--- R i15 : diff(matrix {{x},{y}}, matrix {{x^2, x*y, y^2}}) o15 = | 2x y 0 | | 0 x 2y | 2 3 o15 : Matrix R <--- R i16 : v = matrix {{x,y}} o16 = | x y | 1 2 o16 : Matrix R <--- R i17 : diff(v ** transpose v, 3*x^2 + 5*x*y + 11*y^2) o17 = {1} | 6 5 | {1} | 5 22 | 2 2 o17 : Matrix R <--- R i18 : f = x^3 + y^3 + z^3 - t*x*y*z 3 3 3 o18 = - t*x*y*z + x + y + z o18 : R i19 : v = matrix {{x,y,z}} o19 = | x y z | 1 3 o19 : Matrix R <--- R i20 : det diff(transpose v * v, f) 3 2 3 2 3 2 3 o20 = - 2t x*y*z - 6t x - 6t y - 6t z + 216x*y*z o20 : R i21 : contract(x,m) o21 = | x2 x3 | | x4 x5 | 2 2 o21 : Matrix R <--- R i22 : contract(x^2,m) o22 = | x x2 | | x3 x4 | 2 2 o22 : Matrix R <--- R i23 : contract(matrix {{x,x^2,x^3,x^4}}, m) o23 = | x2 x3 x x2 1 x 0 1 | | x4 x5 x3 x4 x2 x3 x x2 | 2 8 o23 : Matrix R <--- R i24 : contract(matrix {{x,x^2},{x^3,x^4}}, m) o24 = | x2 x3 x x2 | | x4 x5 x3 x4 | | 1 x 0 1 | | x2 x3 x x2 | 4 4 o24 : Matrix R <--- R i25 : f 3 3 3 o25 = - t*x*y*z + x + y + z o25 : R i26 : v3 = symmetricPower(3,matrix{{x,y,z}}) o26 = | x3 x2y x2z xy2 xyz xz2 y3 y2z yz2 z3 | 1 10 o26 : Matrix R <--- R i27 : contract(v3, f) o27 = | 1 0 0 0 -t 0 1 0 0 1 | 1 10 o27 : Matrix R <--- R i28 : f = a * x^3 + b * x^2 * y + y^3 3 2 3 o28 = a*x + b*x y + y o28 : R i29 : g = b * x^3 + a * x * y^2 + y^3 3 2 3 o29 = b*x + a*x*y + y o29 : R i30 : n = matrix {{f,g}} ** symmetricPower(2,matrix {{x,y}}) o30 = | ax5+bx4y+x2y3 ax4y+bx3y2+xy4 ax3y2+bx2y3+y5 bx5+ax3y2+x2y3 ----------------------------------------------------------------------- bx4y+ax2y3+xy4 bx3y2+axy4+y5 | 1 6 o30 : Matrix R <--- R i31 : M = contract(transpose symmetricPower(5,matrix {{x,y}}), n) o31 = {5} | a 0 0 b 0 0 | {5} | b a 0 0 b 0 | {5} | 0 b a a 0 b | {5} | 1 0 b 1 a 0 | {5} | 0 1 0 0 1 a | {5} | 0 0 1 0 0 1 | 6 6 o31 : Matrix R <--- R i32 : det M 5 2 3 3 2 2 3 4 3 2 2 3 o32 = - a - a b - a b - a b + 2a*b - b + a - 3a b + 3a*b - b o32 : R i33 : diff'(m, matrix {{x,x^2,x^3,x^4}}) o33 = | 3x2 6x 6 0 4x3 12x2 24x 24 | | 5x4 20x3 60x2 120x 6x5 30x4 120x3 360x2 | 2 8 o33 : Matrix R <--- R i34 : diff'(m, matrix {{x,x^2},{x^3,x^4}}) o34 = | 3x2 6x 4x3 12x2 | | 6 0 24x 24 | | 5x4 20x3 6x5 30x4 | | 60x2 120x 120x3 360x2 | 4 4 o34 : Matrix R <--- R i35 : contract'(m, matrix {{x,x^2,x^3,x^4}}) o35 = | x2 x 1 0 x3 x2 x 1 | | x4 x3 x2 x x5 x4 x3 x2 | 2 8 o35 : Matrix R <--- R i36 : contract'(m, matrix {{x,x^2},{x^3,x^4}}) o36 = | x2 x x3 x2 | | 1 0 x 1 | | x4 x3 x5 x4 | | x2 x x3 x2 | 4 4 o36 : Matrix R <--- R i37 :