Applied Mathematics Notes | B.Sc. (Information Technology) Semester 3 | Mumbai University | munotes

Contents

Module I 1 Complex Numbers: Complex number, Equality of complex numbers, Graphical representation of complex number (Argand's Diagram), Polar form

1. What This Subject Is About, and Where an IT Student Meets It 1
2. How This Paper Is Examined, and What That Means for Your Revision 3
3. Why the Square Root of Minus One Had To Be Invented 5
4. The Complex Number, and Its Two Parts 7
5. The Powers of i, and the Cycle of Four 10
6. Equality of Two Complex Numbers, and the Two Equations It Gives You 12
7. Adding and Subtracting Complex Numbers 15
8. Multiplying Complex Numbers 17
9. The Conjugate, and the Three Things It Is For 19
10. Dividing Complex Numbers 22
11. The Argand Diagram: Drawing a Complex Number 24
12. The Modulus, and Distance in the Plane 27
13. The Argument, and the Trap Every Calculator Sets 30
14. Addition and Subtraction on the Argand Diagram 33
15. Multiplication and Division on the Argand Diagram 36
16. The Polar Form of a Complex Number 38
17. The Polar Form of x + iy for Every Combination of Signs 40
18. Multiplying and Dividing in Polar Form 42
19. De Moivre's Theorem 44
20. Taking a Power with De Moivre's Theorem 47
21. The n Roots of a Complex Number 49
22. The Roots of Unity, and Why a Computer Scientist Meets Them 52
23. Euler's Formula, and the Exponential Form 54
24. Why e to the i theta Really Is cos theta + i sin theta 56
25. The Exponential Form at Work 58
26. The Logarithm of a Complex Number 60
27. Circular Functions of a Complex Angle 62
28. Separating sin(x + iy) and cos(x + iy) 64
29. The Hyperbolic Functions 66
30. The Hyperbolic Identities 68
31. The Relations Between the Circular and the Hyperbolic Functions 70
32. Osborn's Rule, and Where It Fails 72
33. tan(x + iy), and Separating It Into Real and Imaginary Parts 74
34. The Inverse Hyperbolic Functions, and Why They Are Logarithms 76
35. Inverse Circular Functions of a Complex Number 78
36. Separating an Inverse Function Into Real and Imaginary Parts 80
37. Indeterminate Forms, and Why Zero Over Zero Is Not a Number 82
38. L'Hopital's Rule, and the Three Places This Paper Needs It 84
39. What a Transform Is, and Why Anyone Would Want One 86
40. The Definition of the Laplace Transform 88
41. Does the Integral Exist? Piecewise Continuity and Exponential Order 91
42. The Transform of 1, of t, and of t to the n, From the Definition 94
43. The Transform of e to the at, From the Definition 96
44. The Transform of sin at and cos at, From the Definition 98
45. The Transform of sinh at and cosh at 100
46. The Table of Elementary Transforms 102
47. Linearity, and What It Does and Does Not Let You Do 104
48. The First Shifting Theorem 106
49. The First Shifting Theorem at Work 108
50. Multiplying by t: Differentiating the Transform 110
51. Multiplying by a Power of t 112
52. Dividing by t: Integrating the Transform 114
53. Change of Scale 116
54. The Transform of a Derivative 118
55. The Transform of an Integral 120
56. The Initial and Final Value Theorems 122
57. The Unit Step Function 125
58. The Second Shifting Theorem 127
59. Writing a Piecewise Function With Step Functions 129
60. The Transform of a Periodic Function 132
61. The Unit Impulse, and the Dirac Delta Function 134
62. The Transform of Special Functions 137
63. The Convolution Theorem 139
64. Proving the Convolution Theorem, and Using It Forwards 142
65. What the Inverse Transform Is 145
66. The Table Read Backwards 147
67. Completing the Square, and the First Shifting Theorem Backwards 149
68. Partial Fractions: Distinct Linear Factors 151
69. Partial Fractions: Repeated Factors 153
70. Partial Fractions: Irreducible Quadratic Factors 155
71. The Cover-Up Rule, and How to Check a Partial Fraction in Ten Seconds 157
72. Inverting With the Second Shifting Theorem 160
73. Inverting by Convolution 162
74. Inverting When F(s) Is a Derivative, an Integral, or Has a Factor of s 164
75. Solving a Differential Equation by the Transform: The Method 166
76. Worked First-Order Initial Value Problems 168
77. Second-Order Initial Value Problems 170
78. Equations Driven by a Step or an Impulse 172
79. Simultaneous Differential Equations by the Transform 174

Module II 1 Equation of the first order and of the first degree: Separation of variables, 15 Hrs

1. What a Differential Equation Is 176
2. Order, Degree, and Linearity 178
3. Where Differential Equations Come From 180
4. Forming a Differential Equation by Eliminating the Constants 182
5. What a Solution Is: General, Particular, and Singular 184
6. Separating the Variables 187
7. Worked Separable Equations 189
8. Equations That Become Separable by a Substitution 191
9. Equations Homogeneous in x and y 193
10. Worked Homogeneous Equations 196
11. Non-Homogeneous Linear Equations 199
12. When the Two Lines Are Parallel 201
13. The Exact Differential Equation 203
14. The Test for Exactness, and Why It Works 205
15. Solving an Exact Equation 207
16. Integrating Factors: What They Are and Why They Exist 210
17. Finding an Integrating Factor by Inspection 212
18. The Integrating Factor of a Homogeneous Equation 214
19. The Integrating Factor When the Equation Has the Form y f(xy) dx + x g(xy) dy 216
20. The Integrating Factor That Depends on x Alone 218
21. The Integrating Factor That Depends on y Alone 220
22. The Linear Equation of the First Order 223
23. Bernoulli's Equation, and Other Equations Reducible to the Linear Form 226
24. The Method of Substitution, Gathered Into One Place 229
25. First-Order Equations in Practice: Decay, Cooling, Circuits and Trajectories 231
26. An Equation of the First Order and a Degree Higher Than the First 234
27. Solvable for p: The Method of Factors 236
28. Worked Equations Solvable for p 239
29. Solvable for y 241
30. Worked Equations Solvable for y 243
31. Solvable for x 245
32. Clairaut's Form 247
33. The Singular Solution, and the Envelope 249
34. Reducing an Equation to Clairaut's Form 252
35. Choosing the Method: A Decision Table for Module 2 254
36. What a Linear Equation With Constant Coefficients Is 256
37. The Differential Operator D 259
38. The Laws D Obeys, and the One It Does Not 261
39. The Auxiliary Equation 263
40. Case One: Real and Distinct Roots 265
41. Case Two: Repeated Roots 267
42. Case Three: Complex Roots 269
43. Case Four: Repeated Complex Roots 271
44. The Complete Solution: Complementary Function Plus Particular Integral 273
45. The Complementary Function, and What It Physically Means 276
46. The Inverse Operator 1 Over f(D) 278
47. The Particular Integral When X Is an Exponential 280
48. When the Exponential Rule Fails, and What To Do 282
49. The Particular Integral When X Is a Sine or a Cosine 284
50. When the Sine Rule Fails 286
51. The Particular Integral When X Is a Polynomial 288
52. The Particular Integral When X Is an Exponential Times Something Else 290
53. The Particular Integral When X Is x Times a Function 293
54. The General Method, For When No Rule Fits 295
55. Worked Equations End to End 298
56. Where These Equations Come From: Circuits, Springs and Signals 301