1. Acknowledgements
2. Introduction
3. PART I
4. The basics - derivation and integration
5. Derivatives as rate of change
6. Integrals as the area under the curve
7. Derivatives and integrals of functions are also functions
8. An unexpected relationship
9. Time to use all that
10. The tangent problem
11. The constant function
12. The linear function f(x)=ax
13. The polynomial function f(x)=xa
14. The exponential function f(x)=ex
15. The general exponential function f(x)=ax
16. The natural logarithm function f(x)=ln(x)
17. The derivatives of f(x)=sin(x) and f(x)=cos(x)
18. Some takeaways so far
19. Now we are getting serious - properties of derivation and integration
20. Properties of derivation
21. Properties of integrals
22. We have the tools, let's complicate it a bit
23. Integrals and derivatives in the real world
24. A few more takeaways
25. The universe isn't a line: vector and multivariable Calculus
26. Hey, I'm walking here!
27. Representing vectors
28. Starting simple: vector fields of one variable
29. Next: vector fields of multiple variables
30. Partial derivatives, one variable at a time
31. Derivatives as the tangent plane
32. More of the same: maximum of a surface
33. Notation, notation, notation...
34. What about properties?
35. Intermission: before we move any further, let's talk linear operators
36. The Four Knights of Linear Algebra
37. Takeaways from partial derivatives
38. Ok, I get partial derivatives, what about integrals?
39. Integrating in two directions: the volume under the surface
40. Change of variables in an integral: the Jacobian
41. The general definition
42. Takeaways from this one
43. Nice, but it feels like there's more to it...
44. Step 1: recognize this is no longer an integral of two variables
45. Step 2: realize ds can be thought as a small straight segment of the curve C
46. Step 3: replace these equations in the original integral
47. Examples
48. Takeaways from this section
49. Modeling the Universe: differential equations
50. But first, the basics: the linear ordinary differential equation - LODE
51. First-order linear ordinary differential equation
52. The second-order ordinary differential equation
53. Higher order ordinary differential equations
54. Ok, nice, but why differential equations for modeling systems?
55. Turning the heat up: partial differential equations - PDEs
56. Common PDEs modeling different systems
57. And to solve these equations...
58. Takeaways from this section
59. The final frontier: triple and surface integrals
60. If the double integral is the volume under a surface, the triple integral is...what?
61. What about the surface integral?
62. End of part I
63. PART II
64. Using Calculus in real life
65. Angles and distances: polar coordinates
66. Gravity: of orbiting planets and falling apples
67. Joseph Fourier and heat: the Fourier series and Fourier transform
68. From functions to lines: linearizing is the key
69. Controlling systems with Laplace's transform
70. From radio to relativity: Maxwell's waves
71. The triad of electronics: resistors, capacitors and inductors
72. Are Newton's methods too complicated? Lagrangian mechanics to the rescue
73. Of cats and probability waves: Schrödinger's equations
74. The world is radioactive, including you
75. Beating Earth's gravity: the rocket equation
76. The Big Bang Theory show: the actual math behind Feynman's trick
77. That's all, folks
78. Pendulum on a movable support
79. Hydrogen atom
80. The wave guide
