#1: " ----------[ D e r i v e ]---------- " User #2: " ---------- Initialization ---------- " User #3: " ---------- Vector Analysis ---------- " User #4: " Vector norm => sqrt(15) " User #5: |[1 + #i, -2, 3*#i]| User #6: SQRT(15) Simp(#5) #7: " Cross product: (2, 2, -3) x (1, 3, 1) => (11, -5, 4) "User #8: CROSS([2, 2, -3], [1, 3, 1]) User #9: [11, -5, 4] Simp(#8) #10: a :epsilon Nonscalar User #11: a Simp(#10) #12: b :epsilon Nonscalar User #13: b Simp(#12) #14: c :epsilon Nonscalar User #15: c Simp(#14) #16: d :epsilon Nonscalar User #17: d Simp(#16) #18: f :epsilon Nonscalar User #19: f Simp(#18) #20: g :epsilon Nonscalar User #21: g Simp(#20) User #22: " (a x b) . (c x d) => (a . c) (b . d) - (a . d) (b . c) " #23: CROSS(a, b) . CROSS(c, d) User #24: CROSS(a, b) . CROSS(c, d) Simp(#23) User #25: " => (2 y z^3 - 2 x^2 y^2 z, x y, 2 x y^2 z^2 - x z) " / 2 2 2 2 3\ #26: CURL(\x*y*z, x *y *z , y *z /) User / 3 2 2 2 2 \ #27: \2*y*z - 2*x *y *z, x*y, x*(2*y *z - z)/ Simp(#26) #28: " DEL . (f x g) => g . (DEL x f) - f . (DEL x g) " User #29: DIV(CROSS(f, g)) User / / x y z \\ #30: DIV|CROSS(f, g), | || Simp(#29) \ \ 1 1 1 // #31: a := User #32: b := User #33: c := User #34: d := User #35: f := User #36: g := User User #37: " Express DEL . a in spherical coordinates (r, theta, phi) for " User #38: " a = (a_r(r, theta, phi), a_theta(r, theta, phi), a_phi(r, theta, phi)) " User #39: " Here, phi is in the x-y plane and theta is the angle with the z-axis. " User #40: " (Note that in Derive, the roles of theta and phi are switched.) " User #41: " => 1/r^2 d/dr[r^2 a_r] + 1/[r sin(theta)] d/dtheta[sin(theta) a_theta] " #42: " + 1/[r sin(theta)] da_phi/dphi " User User #43: " => da_r/dr + (2 a_r)/r + 1/r da_theta/dtheta + a_theta/[r tan(theta)] " #44: " + 1/[r sin(theta)] da_phi/dphi " User User #45: " See Keith R. Symon, _Mechanics_, Third Edition, Addison-Wesley Publishing " #46: " Company, 1971, p. 103. " User #47: A_R(r, theta, phi) := User #48: A_THETA(r, theta, phi) := User #49: A_PHI(r, theta, phi) := User User #50: DIV([A_R(r, theta, phi), A_THETA(r, theta, phi), A_PHI(r, theta, phi)], spherical) Simp(#50) #51: COT(phi)*A_PHI(r, theta, phi) ------------------------------- + r d ------- A_THETA(r, theta, phi) d theta -------------------------------- + r*SIN(phi) d d ~ 2*A_R(r, theta, phi) + ----- A_PHI(r, theta, phi) + r*-- A_R~ d phi dr ~ -------------------------------------------------------------~ r ~ (r, theta, phi) ---------------- #52: a_r := User #53: a_theta := User #54: a_phi := User User #55: " Express dR/dt in spherical coordinates (r, theta, phi) where R is the " User #56: " position vector r*Rhat(theta, phi) with Rhat being the unit vector in the " User #57: " direction of R => (dr/dt, r dtheta/dt, r sin(theta) dphi/dt) " #58: " [Symon, p. 98] " User #59: R(t) := User #60: theta(t) := User #61: phi(t) := User #62: RHAT(theta, phi) := User d #63: -- [R(t)*RHAT(theta(t), phi(t)), 0, 0] User dt Simp(#63) / / d #64: |R'(t)*RHAT(theta(t), phi(t)) + R(t)*|phi'(t)* lim -- \ \ @->phi(t) d@ d \ RHAT(theta(t), @) + theta'(t)* lim -- RHAT(@, phi(t))|, @->theta(t) d@ / \ 0, 0| / #65: r := User #66: theta := User #67: phi := User #68: rhat := User #69: " Scalar potential => x^2 y + y + 2 z^3 " User / 2 2\ #70: POTENTIAL(\2*x*y, x + 1, 6*z /) User 2 3 #71: x *y + y + 2*z Simp(#70) User #72: " Vector potential => (x y z, x^2 y^2 z^2, y^2 z^3) is one possible solution. " User #73: " See Harry F. Davis and Arthur David Snider, _Introduction to Vector " User #74: " Analysis_, Third Edition, Allyn and Bacon, Inc., 1975, p. 97. " User / 3 2 2 2 2 #75: q_ := VECTOR_POTENTIAL(\2*y*z - 2*x *y *z, x*y, 2*x*y *z - \ x*z/) #76: q_ User / 2 2 2 \ | y*z *(2*x *y - z ) | #77: |x*y*z, --------------------, 0| Simp(#76) \ 2 / #78: CURL(q_) User / 2 2 2 2 \ #79: \2*y*z*(z - x *y), x*y, x*(2*y *z - z)/ Simp(#78) User #80: " Orthogonalize the following vectors (Gram-Schmidt). See Lee W. Johnson and " User #81: " R. Dean Riess, _Introduction to Linear Algebra_, Addison-Wesley Publishing " User #82: " Company, 1981, p. 104 => [[0 1 2 1], [0 -1 1 -1], [2 1 0 -1]]^T " / 0 1 2 1 \ | | | 0 1 3 1 | #83: q_ := | | User | 1 1 1 0 | | | \ 1 3 6 2 / #84: ORTHONORMAL(q_) User / 1 SQRT(2) \ | 0 --- - --------- ? | | 2 2 | | | | 1 SQRT(2) | | 0 --- --------- ? | | 2 2 | #85: | | Simp(#84) | SQRT(2) 1 | | --------- - --- 0 ? | | 2 2 | | | | SQRT(2) 1 | | --------- --- 0 ? | \ 2 2 / #86: " ---------- Quit ---------- " User