def test_derivatives_in_spherical_coordinates(): GA_Printer.on() X = (r, th, phi) = symbols('r theta phi') curv = [[r * cos(phi) * sin(th), r * sin(phi) * sin(th), r * cos(th)], [1, r, r * sin(th)]] (er, eth, ephi, grad) = MV.setup('e_r e_theta e_phi', metric='[1,1,1]', coords=X, curv=curv) f = MV('f', 'scalar', fct=True) A = MV('A', 'vector', fct=True) B = MV('B', 'grade2', fct=True) assert str(f) == 'f' assert str(A) == 'A__r*e_r + A__theta*e_theta + A__phi*e_phi' assert str( B ) == 'B__rtheta*e_r^e_theta + B__rphi*e_r^e_phi + B__thetaphi*e_theta^e_phi' assert str( grad * f) == 'D{r}f*e_r + D{theta}f/r*e_theta + D{phi}f/(r*sin(theta))*e_phi' assert str( grad | A ) == 'D{r}A__r + 2*A__r/r + A__theta*cos(theta)/(r*sin(theta)) + D{theta}A__theta/r + D{phi}A__phi/(r*sin(theta))' assert str( -MV.I * (grad ^ A) ) == '((A__phi*cos(theta)/sin(theta) + D{theta}A__phi - D{phi}A__theta/sin(theta))/r)*e_r + (-D{r}A__phi - A__phi/r + D{phi}A__r/(r*sin(theta)))*e_theta + (D{r}A__theta + A__theta/r - D{theta}A__r/r)*e_phi' assert str( grad ^ B ) == '(D{r}B__thetaphi - B__rphi*cos(theta)/(r*sin(theta)) + 2*B__thetaphi/r - D{theta}B__rphi/r + D{phi}B__rtheta/(r*sin(theta)))*e_r^e_theta^e_phi' GA_Printer.off() return
def test_derivatives_in_rectangular_coordinates(): GA_Printer.on() X = (x, y, z) = symbols('x y z') (ex, ey, ez, grad) = MV.setup('e_x e_y e_z', metric='[1,1,1]', coords=X) f = MV('f', 'scalar', fct=True) A = MV('A', 'vector', fct=True) B = MV('B', 'grade2', fct=True) C = MV('C', 'mv', fct=True) assert str(f) == 'f' assert str(A) == 'A__x*e_x + A__y*e_y + A__z*e_z' assert str(B) == 'B__xy*e_x^e_y + B__xz*e_x^e_z + B__yz*e_y^e_z' assert str( C ) == 'C + C__x*e_x + C__y*e_y + C__z*e_z + C__xy*e_x^e_y + C__xz*e_x^e_z + C__yz*e_y^e_z + C__xyz*e_x^e_y^e_z' assert str(grad * f) == 'D{x}f*e_x + D{y}f*e_y + D{z}f*e_z' assert str(grad | A) == 'D{x}A__x + D{y}A__y + D{z}A__z' assert str( grad * A ) == 'D{x}A__x + D{y}A__y + D{z}A__z + (-D{y}A__x + D{x}A__y)*e_x^e_y + (-D{z}A__x + D{x}A__z)*e_x^e_z + (-D{z}A__y + D{y}A__z)*e_y^e_z' assert str( -MV.I * (grad ^ A) ) == '(-D{z}A__y + D{y}A__z)*e_x + (D{z}A__x - D{x}A__z)*e_y + (-D{y}A__x + D{x}A__y)*e_z' assert str( grad * B ) == '(-(D{y}B__xy + D{z}B__xz))*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z + (D{z}B__xy - D{y}B__xz + D{x}B__yz)*e_x^e_y^e_z' assert str(grad ^ B) == '(D{z}B__xy - D{y}B__xz + D{x}B__yz)*e_x^e_y^e_z' assert str( grad | B ) == '(-(D{y}B__xy + D{z}B__xz))*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z' assert str(grad < A) == 'D{x}A__x + D{y}A__y + D{z}A__z' assert str(grad > A) == 'D{x}A__x + D{y}A__y + D{z}A__z' assert str( grad < B ) == '(-(D{y}B__xy + D{z}B__xz))*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z' assert str(grad > B) == '0' assert str( grad < C ) == 'D{x}C__x + D{y}C__y + D{z}C__z + (-(D{y}C__xy + D{z}C__xz))*e_x + (D{x}C__xy - D{z}C__yz)*e_y + (D{x}C__xz + D{y}C__yz)*e_z + D{z}C__xyz*e_x^e_y - D{y}C__xyz*e_x^e_z + D{x}C__xyz*e_y^e_z' assert str( grad > C ) == 'D{x}C__x + D{y}C__y + D{z}C__z + D{x}C*e_x + D{y}C*e_y + D{z}C*e_z' GA_Printer.off() return
def make_vector(a, n=3): if isinstance(a, str): sym_str = '' for i in range(n): sym_str += a + str(i + 1) + ' ' sym_lst = list(symbols(sym_str)) sym_lst.append(ZERO) sym_lst.append(ZERO) a = MV(sym_lst, 'vector') return (F(a))
def test_basic_multivector_operations(): GA_Printer.on() (ex, ey, ez) = MV.setup('e*x|y|z') A = MV('A', 'mv') assert str( A ) == 'A + A__x*e_x + A__y*e_y + A__z*e_z + A__xy*e_x^e_y + A__xz*e_x^e_z + A__yz*e_y^e_z + A__xyz*e_x^e_y^e_z' assert str( A ) == 'A + A__x*e_x + A__y*e_y + A__z*e_z + A__xy*e_x^e_y + A__xz*e_x^e_z + A__yz*e_y^e_z + A__xyz*e_x^e_y^e_z' assert str( A ) == 'A + A__x*e_x + A__y*e_y + A__z*e_z + A__xy*e_x^e_y + A__xz*e_x^e_z + A__yz*e_y^e_z + A__xyz*e_x^e_y^e_z' X = MV('X', 'vector') Y = MV('Y', 'vector') assert str(X) == 'X__x*e_x + X__y*e_y + X__z*e_z' assert str(Y) == 'Y__x*e_x + Y__y*e_y + Y__z*e_z' assert str( (X * Y) ) == '(e_x.e_x)*X__x*Y__x + (e_x.e_y)*X__x*Y__y + (e_x.e_y)*X__y*Y__x + (e_x.e_z)*X__x*Y__z + (e_x.e_z)*X__z*Y__x + (e_y.e_y)*X__y*Y__y + (e_y.e_z)*X__y*Y__z + (e_y.e_z)*X__z*Y__y + (e_z.e_z)*X__z*Y__z + (X__x*Y__y - X__y*Y__x)*e_x^e_y + (X__x*Y__z - X__z*Y__x)*e_x^e_z + (X__y*Y__z - X__z*Y__y)*e_y^e_z' assert str( (X ^ Y) ) == '(X__x*Y__y - X__y*Y__x)*e_x^e_y + (X__x*Y__z - X__z*Y__x)*e_x^e_z + (X__y*Y__z - X__z*Y__y)*e_y^e_z' assert str( (X | Y) ) == '(e_x.e_x)*X__x*Y__x + (e_x.e_y)*X__x*Y__y + (e_x.e_y)*X__y*Y__x + (e_x.e_z)*X__x*Y__z + (e_x.e_z)*X__z*Y__x + (e_y.e_y)*X__y*Y__y + (e_y.e_z)*X__y*Y__z + (e_y.e_z)*X__z*Y__y + (e_z.e_z)*X__z*Y__z' (ex, ey) = MV.setup('e*x|y') X = MV('X', 'vector') A = MV('A', 'spinor') assert str(X) == 'X__x*e_x + X__y*e_y' assert str(A) == 'A + A__xy*e_x^e_y' assert str( (X | A) ) == '(-A__xy*((e_x.e_y)*X__x + (e_y.e_y)*X__y))*e_x + (A__xy*((e_x.e_x)*X__x + (e_x.e_y)*X__y))*e_y' assert str( (X < A) ) == '(-A__xy*((e_x.e_y)*X__x + (e_y.e_y)*X__y))*e_x + (A__xy*((e_x.e_x)*X__x + (e_x.e_y)*X__y))*e_y' assert str( (A > X) ) == '(A__xy*((e_x.e_y)*X__x + (e_y.e_y)*X__y))*e_x + (-A__xy*((e_x.e_x)*X__x + (e_x.e_y)*X__y))*e_y' (ex, ey) = MV.setup('e*x|y', metric='[1,1]') X = MV('X', 'vector') A = MV('A', 'spinor') assert str(X) == 'X__x*e_x + X__y*e_y' assert str(A) == 'A + A__xy*e_x^e_y' assert str( (X * A)) == '(A*X__x - A__xy*X__y)*e_x + (A*X__y + A__xy*X__x)*e_y' assert str((X | A)) == '-A__xy*X__y*e_x + A__xy*X__x*e_y' assert str((X < A)) == '-A__xy*X__y*e_x + A__xy*X__x*e_y' assert str((X > A)) == 'A*X__x*e_x + A*X__y*e_y' assert str( (A * X)) == '(A*X__x + A__xy*X__y)*e_x + (A*X__y - A__xy*X__x)*e_y' assert str((A | X)) == 'A__xy*X__y*e_x - A__xy*X__x*e_y' assert str((A < X)) == 'A*X__x*e_x + A*X__y*e_y' assert str((A > X)) == 'A__xy*X__y*e_x - A__xy*X__x*e_y' GA_Printer.off() return