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mathemagizian
Estracto del artículo
"300 conceptos de mate que deberias saber como un pre-matemático profesional"
en el wordpress.com:
... Elementary maths (nowadays dubbed precalculus) are those that you are learning since the awakening of your conscience
and anyone should master no matter what profession wants to jump in. After you learn those elementary maths that each human mind should learn to perform dilligently in a
competitive and
cruel society, you would be prepared to begin to learn a set of concepts that serve to you to be considered a professional mathematician, a list of concepts and results
is offered to do a contrast against you think you know or should know.
Independently of the flavor or kind of mathematician you gonna be, the following list is aimed for a universal person
which is an aspirant mathematician who claim to be useful in the whatever enviroment where he or she evolves. Remember that as an amateur mathematician you are fated to
be abstract minded and perhaps to apply these techniques to the reality, to model matter and energy as well as all its tangible instances. Abstract reasoning is our
essential spirit.
For a begining professional mathematician an incomplete list of
modern concepts and results is:
- sets and maps
- set operations
- symmetric difference
- basic types of maps: constant, injective, surjective, bijective
- image, preimage, rank and fiber of a mapping
- equivalence relation
- equivalence class
- partitions
- abstraction lemma
- combinations
- binomial theorem
- division algorithm
- Euclides algorithm
- prime numbers
- arithmetic fundamental theorem
- coprime integers
- modular arithmetics
- mathematical logic
- propositional and predicative calculus
- axiomatics
- deduction theorem
- Zorn lemma
- Zermelo-Fraenkel set theory axiomatics
- binary operation in a set
- algebraic structure
- semigroup
- monoide
- group
- ring
- field
- skewfield
- module
- vector space
- algebra
- groupoide
- subgroup
- normal subgroup
- coset
- quotient set
- quotient group or factor group
- group morphisms
- kernel of a morphism
- fundamental theroems on group morphisms
- group action
- equation class
- Lagrange theorem
- symmetric group
- permutations
- Cayley theorem
- Cauchy theorem on finite groups
- Sylow theorems
- estructure theorem on abelian groups
- composition series
- subrings and ideals of rings
- algebra of ideals
- factor ring
- ring morphisms
- chinesse remainder theorem
- integral domains
- principal ideal domains
- Euclidean rings
- prime and maximal ideals
- local rings
- unique factorization domains
- radical
- nilradical
- nilpotents
- polynomial rings
- subfields
- field extension
- Galois fields
- Galois correspondence
- Abel theorem
- quaternions
- submodules
- module morphisms
- noetherian modules
- linear combinations
- linear dependence
- linear transformations
- space of linear transformations inter vector spaces
- linear functional or covectors
- matrices
- linear equations
- eigen systems
- change of basis
- determinants
- Gram Schimdt
- orthonormal and unitary matrices
- inner product spaces
- Hamilton Calyley theorem
- Jordan canonical forms
- minimal polynomial
- rational canonical form
- bilinear maps
- quadratic forms
- euclidean geometry
- Pitagoras theorem
- Desargues theorem
- Hilbert axioms
- parallel axioms
- real numbers
- euclidean norm
- $latex \varepsilon$-neighbourhood
- bounded sets
- supremum and infimum
- bounded functions
- limits of sequences and limits of functions
- algebra of limits
- convergence $latex \varepsilon-N$ and $latex \epsilon-\delta$
- Cauchy criterion for convergence
- real number completeness
- numerability or countability
- topology of euclidean spaces
- acummulation points
- partial sums and series
- Hadamard formula
- tests of convergence
- continuity
- algebra of continuity
- mean value theorem
- types of discontinuity
- uniform continuity
- derivative
- algebra of derivatives
- differentiability
- Rolle theorem
- L'Hopital rule
- Taylor and Mclaurin series
- curvilinear coordinates
- partial derivatives
- gradient
- directional derivative
- divergence
- rotational
- hessian
- extrema: critical points
- critical sets
- Lagrange multipliers
- jacobians
- chain's rule
- inverse function theorem
- implicit function theorem
- Morse functions
- Riemann sums and integral
- fundamental theorem of integral calculus
- change of variable integral theorem
- Fubini theorem
- Lebesgue integral
- bounded variation
- complex number
- complex functions
- Cauchy-Goursat theorem
- residue theorem
- Cauchy integral formula
- Cauchy Riemann equation
- Harmonic functions
- winding number
- Laurent series
- Rouche theorem
- maxium modulus theorem
- conformal maps
- Riemann mapping theorem
- fractional linear transformations or Möbius transformations
- hyperbolic geometry
- Poincarč model
- ordinary differential equations
- 1st order edos
- initial valued problems
- boundary valued problems
- existence and unicity of solutions
- Laplace transform
- convolution
- 2nd order and special functions
- Bessel functions
- Laguerre, Legendre, Airy special functions
- hypergeometric functions
- partial differential equations
- 2nd order classification
- Cauchy problem
- Sturm Louville method
- Laplace equation
- diffusion equation
- wave equation
- maximum principle
- Green function
- variables separation
- line and surface integrals
- contour integration
- Green, Gauss, Divergence integral theorems
- Stokes theorem
- duality of vector spaces
- dual space
- Riesz representation theorem
- tensor product of linear transformations
- tensor product of vector spaces
- multilinear algebra of inner product spaces
- parameterizations of curves and surfaces
- curvature
- torsion
- Serret Frenet moving frame
- Gauss map or shape operator
- Weingarten map
- covariant standard derivative or Levi-Civita connection
- Gauss equation
- Egregium theorem
- geodesics
- exponential map
- geodesic curvature
- Gauss-Bonnet theorem
- differential forms
- wedge product
- exterior derivative
- Poncarč lemma
- Grassmann (or exterior) algebra of a vector space
- topological structure
- topological space
- open and closed set
- clausure
- relative topology
- homeomorphism
- completeness
- hausdorffness
- connectedness
- compactness
- Urysohn lemma
- compact open topology
- topological classification of surfaces
- homotopy
- fundamental group
- fiber bundle
- tangent space
- tangent bundle
- differential
- Fourier series
- metric spaces
- spaces of bounded sequences
- Cauchy-Schwarz
- Jordan-Holder
- normed and Banach spaces
- Hilbert spaces
- Hahn Banach theorem
- numerical approximation
- interpolation
- splines
- divided differences
- Newton Rapson
- Gauss Seidel
- Runge Kutta
- midpoint rule
- trapezoidal rule
- Simpson rule
- finite element methods pde's
- generating functions
- sample spaces
- conditional probability
- random variables
- discrete and continous distributions
- expectation
- correlation
- moment generating functions
- law of large numbers
- limit theorem
- limiting distributions
- linear model
- point estimation
- interval estimation
- confidence interval
- hypothesis testing
- Lie groups
- special and general linear groups
- orthonormal groups
- unitary groups
- Lie algebras
- topological manifolds
- differentiable manifolds
- cw-complexes
- Euler characteristic
- fundamentals of computer organization
- computer graphics
- PC architecture
- operating systems
- pseudo code
- c-language
- parallel programing
- numerical computation
- Thruston Nielsen classification of autohomeomorphims
- mapping class group
- free groups
- Schreier free subgroup theorem
- free products of groups
- Kurosch subgroup theorem
- amalgamated products
- Bass Serre theory
- linear programming
- simplex algorithm
- minimax
- variational calculus
- Euler-Lagrange equations
- combinatorial phenomena
- Catalan numbers
- vector fields
- tensor fields
- Riemannian metrics
- Minkowski metrics
- spacetimes
- gravitational fields
- connections
- ...