#1: " ----------[ D e r i v e ]---------- " User #2: " ---------- Initialization ---------- " User #3: " ---------- Calculus ---------- " User User #4: " Calculus on a non-smooth (but well defined) function => x/|x| or sign(x) " d #5: -- |x| User dx #6: SIGN(x) Simp(#5) #7: " Calculus on a piecewise defined function " User #8: A(x) := IF(x < 0, -x, x) User #9: " => if x < 0 then -1 else 1 " User d #10: -- A(x) User dx #11: IF(x < 0, -1, 1) Simp(#10) #12: A(x) := - x*CHI(-inf, x, 0) + x*CHI(0, x, inf) User d #13: -- A(x) User dx #14: SIGN(x) Simp(#13) #15: a := User User #16: " Derivative of a piecewise defined function at a point [Herbert Fischer]. " User #17: " f(x) = x^2 - 1 for x = 1 otherwise x^3. f(1) = 0 and f'(1) = 3 " 2 3 #18: F(x) := IF(x = 1, x - 1, x ) User #19: F(1) User #20: 0 Simp(#19) #21: " Substitute x = 1 in the following " User d #22: -- F(x) User dx 2 #23: IF(x = 1, 2*x, 3*x ) Simp(#22) 2 #24: IF(1 = 1, 2*1, 3*1 ) Sub(#23) #25: 2 Simp(#24) #26: f := User #27: " d^n/dx^n(x^n) => n! " User #28: n :epsilon Integer User #29: n Simp(#28) /d \n n #30: |--| x User \dx/ /d \n n #31: |--| x Simp(#30) \dx/ #32: n := User User #33: " Apply the chain rule---this is important for PDEs and many other " #34: " applications => y_xx (x_t)^2 + y_x x_tt " User #35: X(t) := User #36: Y(x) := User /d \2 #37: |--| Y(X(t)) User \dt/ 2 #38: X''(t)*Y'(X(t)) + X'(t) *Y''(X(t)) Simp(#37) #39: x := User #40: y := User #41: " => f(h(x)) dh/dx - f(g(x)) dg/dx " User #42: F(x) := User #43: G(x) := User #44: H(x) := User H(x) / #45: q_ := / F(y) dy User G(x) d #46: -- q_ User dx #47: H'(x)*F(H(x)) - G'(x)*F(G(x)) Simp(#46) #48: f := User #49: g := User #50: h := User User #51: " Exact differential => d(V(P, T)) => dV/dP DP + dV/dT DT " #52: V(p, t) := User #53: DIF(V(p, t)) User #54: DIF(V(p, t)) Simp(#53) #55: v := User User #56: " Implicit differentiation => dy/dx = [1 - y sin(x y)] / [1 + x sin(x y)] " #57: q_ := y = COS(x*y) + x User #58: IMP_DIF(LHS(q_) - RHS(q_), x, y) User 1 - y*SIN(x*y) #59: ---------------- Simp(#58) x*SIN(x*y) + 1 #60: " => 2 (x + y) g'(x^2 + y^2) " User #61: F(x, y) := User #62: G(z) := User d d #63: q_ := -- F(x, y) + -- F(x, y) User dx dy 2 2 #64: F(x, y) := G(x + y ) User #65: q_ User 2 2 #66: (2*x + 2*y)*G'(x + y ) Simp(#65) #67: f := User #68: g := User #69: " Residue => - 9/4 " User #70: "residue((z^3 + 5)/((z^4 - 1)*(z + 1)), z, -1)" User #71: " Differential forms " User User #72: " (2 dx + dz) /\ (3 dx + dy + dz) /\ (dx + dy + 4 dz) => 8 dx /\ dy /\ dz " #73: "(2*dx + dz) ~ (3*dx + dy + dz) ~ (dx + dy + 4*dz)" User #74: " d(3 x^5 dy /\ dz + 5 x y^2 dz /\ dx + 8 z dx /\ dy) "User #75: " => (15 x^4 + 10 x y + 8) dx /\ dy /\ dz " User #76: "d(3*x^5 * dy ~ dz + 5*x*y^2 * dz ~ dx + 8*z * dx ~ dy)"User #77: " => 1 - 3/8 2^(1/3) = 0.5275296 " User #78: "minimize(x^4 - x + 1)" User #79: " => [0, 1] " User User #80: "[minimize(1/(x^2 + y^2 + 1)), maximize(1/(x^2 + y^2 + 1))]" #81: " Minimize on [-1, 1] x [-1, 1]: " User User #82: " => min(a - b - c + d, a - b + c - d, a + b - c - d, a + b + c + d) " User #83: "minimize(a + b*x + c*y + d*x*y, [x = -1..1, y = -1..1])" #84: " => [-1, 1] " User #85: "[minimize(x**2*y**3, [x = -1..1, y = -1..1]), ~" User #86: " maximize(x**2*y**3, [x = -1..1, y = -1..1])]" User User #87: " Linear programming: minimize the objective function z subject to the " User #88: " variables xi being non-negative along with an additional set of constraints. " User #89: " See William R. Smythe, Jr. and Lynwood A. Johnson, _Introduction to Linear " User #90: " Programming, with Applications_, Prentice Hall, Inc., 1966, p. 117: " User #91: " minimize z = 4 x1 - x2 + 2 x3 - 2 x4 => {x1, x2, x3, x4} = {2, 0, 2, 4} " #92: " with zmin = 4 " User User #93: "simplex(-(4*x1 - x2 + 2*x3 - 2*x4), [2*x1 + x2 + x3 + x4 <= 10, ~" User #94: " x1 - 2*x2 - x3 + x4 >= 4, x1 + x2 + 3*x3 - x4 >= 4])" #95: " ---------- Quit ---------- " User