Sat Jul 11 23:33:15 MET DST 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Calculus ---------- -- Calculus on a non-smooth (but well defined) function => x/|x| or sign(x) D(abs(x), x) abs(x) (1) ------ x Type: Expression Integer Time: 0.20 (IN) + 0.30 (EV) + 0.22 (OT) + 0.20 (GC) = 0.92 sec -- Calculus on a piecewise defined function a(x) == if x < 0 then -x else x Type: Void Time: 0 sec -- => if x < 0 then -1 else 1 D(a(x), x) Compiling function a with type Variable x -> Polynomial Integer (3) 1 Type: Polynomial Integer Time: 0.28 (IN) + 0.03 (OT) = 0.32 sec )clear properties a Compiled code for a has been cleared. -- Derivative of a piecewise defined function at a point [Herbert Fischer]. -- f(x) = x^2 - 1 for x = 1 otherwise x^3. f(1) = 0 and f'(1) = 3 f(x) == if x = 1 then x**2 - 1 else x**3 Type: Void Time: 0.02 (IN) = 0.02 sec f(1) Compiling function f with type PositiveInteger -> Integer (5) 0 Type: NonNegativeInteger Time: 0.10 (IN) + 0.02 (EV) = 0.12 sec D(f(x), x) Compiling function f with type Variable x -> Polynomial Integer 2 (6) 3x Type: Polynomial Integer Time: 0.18 (IN) + 0.08 (OT) = 0.27 sec subst(%, x = 1) (7) 3 Type: Expression Integer Time: 0.25 (IN) + 0.03 (EV) + 0.05 (OT) = 0.33 sec )clear properties f Compiled code for f has been cleared. -- d^n/dx^n(x^n) => n! D(x**n, x, n) There are 3 exposed and 0 unexposed library operations named D having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op D to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named D with argument type(s) Expression Integer Variable x Variable n -- Apply the chain rule---this is important for PDEs and many other -- applications => y_xx (x_t)^2 + y_x x_tt x:= operator('x); Type: BasicOperator Time: 0.05 (IN) + 0.02 (OT) = 0.07 sec y:= operator('y); Type: BasicOperator Time: 0.05 (IN) = 0.05 sec D(y(x(t)), t, 2) , 2 ,, , ,, (10) x (t) y (x(t)) + y (x(t))x (t) Type: Expression Integer Time: 0.42 (IN) + 0.05 (EV) + 0.12 (OT) = 0.58 sec )clear properties x y -- => f(h(x)) dh/dx - f(g(x)) dg/dx f:= operator('f); Type: BasicOperator Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec g:= operator('g); Type: BasicOperator Time: 0.05 (IN) = 0.05 sec h:= operator('h); Type: BasicOperator Time: 0.03 (IN) + 0.02 (OT) = 0.05 sec 'integrate(f(y), y = g(x)..h(x)) (14) integrate(f(y),y= g(x),h(x) ) Type: OutputForm Time: 0 sec D(%, x) There are 5 exposed and 0 unexposed library operations named D having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op D to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named D with argument type(s) OutputForm Variable x )clear properties f g h -- Exact differential => d(V(P, T)) => dV/dP DP + dV/dT DT V:= operator('V); Type: BasicOperator Time: 0.03 (IN) + 0.02 (OT) = 0.05 sec D(V(P, T)) There are 2 exposed and 0 unexposed library operations named D having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op D to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named D with argument type(s) Expression Integer )clear properties V -- Implicit differentiation => dy/dx = [1 - y sin(x y)] / [1 + x sin(x y)] y = cos(x*y) + x (16) y= cos(x y) + x Type: Equation Expression Integer Time: 0.22 (IN) + 0.12 (EV) + 0.05 (OT) + 0.02 (GC) = 0.40 sec -- => 2 (x + y) g'(x^2 + y^2) f:= operator('f); Type: BasicOperator Time: 0.02 (OT) = 0.02 sec g:= operator('g); Type: BasicOperator Time: 0.02 (OT) = 0.02 sec D(f(x, y), x) + D(f(x, y), y) (19) f (x,y) + f (x,y) ,2 ,1 Type: Expression Integer Time: 0.05 (IN) + 0.03 (EV) + 0.02 (OT) = 0.10 sec subst(%, f(x, y) = g(x**2 + y**2)) (20) f (x,y) + f (x,y) ,2 ,1 Type: Expression Integer Time: 0.10 (IN) + 0.02 (OT) = 0.12 sec )clear properties f g -- Residue => - 9/4 --residue((z**3 + 5)/((z**4 - 1)*(z + 1)), z, -1) -- Differential forms DeRham:= DERHAM(Integer, [x, y, z]) (21) DeRhamComplex(Integer,[x,y,z]) Type: Domain Time: 0.02 (IN) = 0.02 sec [dx, dy, dz]:= [generator(i)$DeRham for i in 1..3] (22) [dx,dy,dz] Type: List DeRhamComplex(Integer,[x,y,z]) Time: 0.02 (EV) + 0.02 (OT) = 0.03 sec -- (2 dx + dz) /\ (3 dx + dy + dz) /\ (dx + dy + 4 dz) => 8 dx /\ dy /\ dz (2*dx + dz) * (3*dx + dy + dz) * (dx + dy + 4*dz) (23) 8dx dy dz Type: DeRhamComplex(Integer,[x,y,z]) Time: 0.03 (IN) + 0.02 (EV) = 0.05 sec -- d(3 x^5 dy /\ dz + 5 x y^2 dz /\ dx + 8 z dx /\ dy) -- => (15 x^4 + 10 x y + 8) dx /\ dy /\ dz totalDifferential( _ 3*x**5 * dy * dz + 5*x*y**2 * dz * dx + 8*z * dx * dy :: DeRham) Internal Error The function * with signature hashcode is missing from domain Polynomial(DeRhamComplex (Integer) (x y z)) initial (24) -> real 12.1 user 5.6 sys 0.4 ------------------------------------------------------------------------------- Sat Jul 11 23:35:58 MET DST 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Calculus ---------- -- => 1 - 3/8 2^(1/3) = 0.5275296 --minimize(x**4 - x + 1) -- => [0, 1] --[minimize(1/(x**2 + y**2 + 1)), maximize(1/(x**2 + y**2 + 1))] -- Minimize on [-1, 1] x [-1, 1]: -- => min(a - b - c + d, a - b + c - d, a + b - c - d, a + b + c + d) --minimize(a + b*x + c*y + d*x*y, [x = -1..1, y = -1..1]) -- => [-1, 1] --[minimize(x**2*y**3, [x = -1..1, y = -1..1]), _ -- maximize(x**2*y**3, [x = -1..1, y = -1..1])] -- Linear programming: minimize the objective function z subject to the -- variables xi being non-negative along with an additional set of constraints. -- See William R. Smythe, Jr. and Lynwood A. Johnson, _Introduction to Linear -- Programming, with Applications_, Prentice Hall, Inc., 1966, p. 117: -- minimize z = 4 x1 - x2 + 2 x3 - 2 x4 => {x1, x2, x3, x4} = {2, 0, 2, 4} -- with zmin = 4 --simplex(-(4*x1 - x2 + 2*x3 - 2*x4), [2*x1 + x2 + x3 + x4 <= 10, _ -- x1 - 2*x2 - x3 + x4 >= 4, x1 + x2 + 3*x3 - x4 >= 4]) -- ---------- Quit ---------- )quit real 0.9 user 0.7 sys 0.1