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Linear Algebra

1. Introduction to Linear Algebra
2. Matrices
3. Matrix factorizations
4. Linear Systems
5. Spectral decomposition
- 5.1. Spectral decomposition of symmetric matrices
- 5.2. Sensitivity of spectral decomposition
6. Cayley-Hamilton theorem

Multivariable Calculus

7. Introduction to multi-variable calculus
8. Interchanging operators

Differential Geometry

9. Introduction to Differential Geometry

Vector and Tensor Algebra and Calculus

10. Tensor Algebra
11. Tensor Calculus in Euclidean Spaces
12. Tensor Invariants
13. Unitary and rotation tensors
14. Isotropic Tensors
15. Time derivative of integrals over moving domains
16. Calculus identities

Functional Analysis

17. Introduction to Functional Analysis
18. Dirac’s delta

Complex Calculus

19. Complex Algebra
20. Complex Analysis
- 20.5. Sequences and Series in the Complex Field
21. Laplace Transform
- 21.1. Definition and Properties
- 21.2. Applications of Laplace Transform
22. Fourier Transforms
23. Z-transform

Approximation of Functions

24. Introduction to function approximation
25. Lagrange interpolation

Calculus of Variations

26. Introduction to Calculus of Variations

Ordinary Differential Equations

27. Introduction to Ordinary Differential Equations
28. Linear equations
29. Linear systems
30. Integration schemes for Cauchy problems

System Theory

31. Introduction to System Theory
32. Linear Time-Invariant Systems
33. Coordinate transformation
34. LTI system response
35. LTI: stability and feedback
36. Observability and Detectability
37. Reachability and Controllability
38. State-space realizations
39. Kalman decomposition
40. Lyapunov equation
41. Order reduction of dynamical systems
- 41.1. Balanced truncation
42. Shape filter
43. Padé approximation of time delay

Control Theory

44. Introduction to control methods
45. Closed-loop control: requirements and performance
- 45.1. Stability (SISO)
- 45.2. Reference tracking (SISO)
46. Frequency domain control
47. Optimal control
48. Pole placement
49. Examples

Partial Differential Equations

50. Introduction to Partial Differential Equations
51. Elliptic equations
52. Parabolic equations
53. Hyperbolic equations
54. Navier-Cauchy equations
55. Navier-Stokes equations
56. Arbitrary Lagrangian-Eulerian description

Numerical Methods for PDEs

57. Introduction to numerical methods for PDEs
58. Finite Element Method
- 58.1. 1-dimensional Poisson equation
59. Finite Volume Method
60. Boundary Element Method

Boundary Methods for PDEs

61. Green’s function method

Optimization

62. Optimization

Reinforcement Learning

63. Introduction to Reinforcement Learning
64. Markov Processes
65. Methods of solution of MPD: DP and LP
66. Methods of solution of MPD: RL
67. Large or Continuous MDPs

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Finite Volume Method

59. Finite Volume Method#

Property 59.1

Evaluate flux on interfaces between cells, and distribute between neighboring cells.

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58.1. 1-dimensional Poisson equation

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59.1. 1-dimensional Poisson equation

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