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Three complete paths, each starting from definitions.
Foundations of calculus
Limits → Derivative → Mean value theorem → Integral → Taylor expansion
- 1Limits
- 2The derivative
- 3Mean value theorem
- 4The integral
- 5Taylor expansion
5 concepts · 4 experiments · 10 exercises
Foundations of linear algebra
Vectors → Inner products → Linear maps → Matrices → Eigenvalues
- 1Vectors
- 2Inner products & Cauchy–Schwarz
- 3Linear maps
- 4Matrices
- 5Eigenvalues
5 concepts · 1 experiments · 10 exercises
Introduction to differential equations
What is an ODE → First-order → Second-order linear → Numerical solutions
- 1What is an ODE
- 2First-order equations
- 3Second-order linear equations
- 4Numerical solutions
4 concepts · 1 experiments · 8 exercises
- LimitsSaying “arbitrarily close” precisely with ε–δ.
- The derivativeThe limit of a rate of change, and why eˣ is its own derivative.
- Mean value theoremThe overall average rate of change is attained at some instant; Rolle → Lagrange, and why no hypothesis can be dropped.
- The integralThe limit of Riemann sums and the fundamental theorem.
- Taylor expansionApproximating functions by polynomials, with error control.
- VectorsAxioms of a vector space; bases and coordinates.
- Inner products & Cauchy–SchwarzWhere length and angle come from; the whole inequality follows from “a projection is never longer”.
- Linear mapsMaps preserving addition and scaling, determined entirely by what they do to a basis.
- MatricesCoordinate representation of linear maps; why matrix multiplication is defined the way it is.
- EigenvaluesDirections along which a map is just a stretch.
- What is an ODEDescribing change through derivatives; existence and uniqueness of solutions.
- First-order equationsSeparation of variables, integrating factors and the geometry of solutions.
- Second-order linear equationsWhy the characteristic equation works: it is an eigenvalue problem.
- Numerical solutionsEuler and RK4: step size, error and the connection to Taylor expansion.