Applied III woksheet (original) (raw)

Civil All Formulas

Mathematics 1. If ≠ ′, no solution, if = ′ = , unique solution if = ′ < , many solutions. (non-homogeneous) 2. If = , trivial solution, if < ,then (−) linearly independent solutions. (Many solutions) and if < , then many solutions. 3. (+ ℎ) = () + ℎ ′ () + ℎ 2 2! ′′ () + ℎ 3 3! ′′′ () + … … … ∞ 4. If − 2 > 0 and < 0 (,) have maximum, if − 2 > 0 and > 0 (,) have minimum at(,) and if − 2 < 0, then saddle point. If − 2 = 0, ℎ investigation is required to decide. 5. ∫ (∅ +) = ∫ ∫ (− ∅) (Green's) 6. ∫. ℝ=∫. (Stokes) 7. ∫. = ∫ (Gauss) 8. ∫ = ∫ ∫ + 9. If + = 0 be a homogeneous equation in and , then 1 + an integrating factor 10. If the equation of the type 1 () + 2 () = 0. If the equation + = 0 be of this type then 1 − is an integrating factor 11. If − be a function of x only = () say then ∫ () is an integrating factor 12. If − be a function of y only = () say then ∫ () is an integrating factor. 13. ∫ = + ∫ terms of N not containing x dy = c 14.. . = 1 () = 1 () , () ≠ 0, if () = 0, ℎ. . = 1 ′ () , ′ () ≠ 0 15.. = 1 (2) sin(+) = 1 (− 2) , (− 2) ≠ 0, if (− 2) = 0, then. . = 1 ′ (− 2) sin(+), ′ (− 2) ≠ 0 16.. . = 1 () = 1 (+) 17.. = 1 () = [ ()] −1 , 18. (1 +) −1 = 1 − + 2 − ⋯ 19. (1 −) −1 = 1 + + 2 + ⋯ 20. + 1 −1 −1 −1 + ⋯ −1 + = , = , = , 2 2 2 = (− 1) , 3 3 3 = D(D − 1)(D − 2) 21. (∫ ℎ(,) () ()) = ∫ ℎ(,) () () + ℎ[ (), ] − ℎ[ (), ] 22. { ()} = ∫ − ∞ 0 () 23. (1) = 1 24. () = ! +1 25. () = 1 − 26. (sin) = 2 + 2 27. (cos) = 2 + 28. (sinh) = 2 − 2 29. (cosh) = 2 − 2 30. { ()} = ̅ (−) 31. (+) = () then { ()} = ∫ − () 0 1− − 32. { ′ ()} = ̅ () − (0) 33. { ()} = ̅ () − −1 (0) − −2 ′ (0) − ⋯ … …. . −1 (0) 34. {∫ () 0 } = 1 ̅ () 35. { ()} = (−1). [ ̅ (s)] 36. { 1 ()} = ∫ ̅ (s) ∞ 37. () = 0 2 + ∑ ∞ =1 cos + ∑ ∞ =1 sin 38. 0 = 1 ∫ () +2 , = 1 ∫ () cos +2 , = 1 ∫ () sin +2 39. () = 0 2 + ∑ ∞ =1 cos + ∑ ∞ =1 sin 40. 0 = 1 ∫ () +2 , = 1 ∫ () cos +2 , = 1 ∫ () sin +2 41. () = ∑ ∞ =1 sin , where = 2 ∫ () sin 0 42. () = 0 2 + ∑ ∞ =1 cos where, 0 = 2 ∫ () 0 , = 2 ∫ () cos 0 43. = ∑ () = ∫ () ∞ −∞ 44. 2 = ∑ (−) 2 () 2 = ∫ (−) 2 () ∞ −∞ 45. : = : 2 = (Poisson's distribution) 46. () = 1 √2 − 1 2 (−) 2 (Normal distribution) 47. = + , ∑ = + ∑ , ∑ = ∑ + ∑ 2 48. 50. () 0 = 1 ℎ [∆ 0 − 1 2 ∆ 2 0 + 1 3 ∆ 3 0 − 1 4 ∆ 4 0 + ⋯ ] 51. () = 1 ℎ [∇ + 1 2 ∇ 2 + 1 3 ∇ 3 + 1 4 ∇ 4 + ⋯ ] 52. +1 = − () ′ () (Newton-Raphson) 53. ∫ () 0 + ℎ 0 = ℎ 2 [ 0 + + 2(1 + 2 + ⋯. . + −1)] (Trapezoidal) 54. ∫ () 0 + ℎ 0 = ℎ 3 [(0 +) + 4(1 + 3 + ⋯ −1) + 2(2 + 4 + ⋯ −2)] (Simpson's) 55. = − − 12 ℎ 2 ′′ () = (ℎ 2) (Trapezoidal) 56. = − − 180 ℎ 4 () = (ℎ 4) (Simpson's) 57. +1 = + ℎ. (,) where = (,) (Euler's) 2

Chapter 2. Technical Mathematics Signed Numbers

2-29. (a)-6 0 C; (b)-17 0 C; (c) 36 0 C 2-30. L = 2 mm[(-30 0 C) – (-5 0 C)] = 2 mm(-25) =-50 mm; Decrease in length. Algebra Review 2-31. x = (2) + (-3) + (-2) =-3; x =-3 2-32. x = (2) – (-3) – (-2) = +7; x = +7 2-33. x = (-3) + (-2)-(+2) =-7; x =-7 2-34. x =-3[(2) – (-2)] =-3(2 + 2) =-12; x =-12