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131: MathematicalSpaces

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mind map of 131 concepts for fall 17 final
Ruby Kalra
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Ruby Kalra
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Zusammenfassung der Ressource

131: MathematicalSpaces
  1. Linear Spaces (Vector Spaces)
    1. Vector Subspaces
      1. contain elements
        1. infinite dimensional
          1. functions
            1. real
              1. defining properties
              2. complex
            2. finite dimensional
              1. vectors
                1. real
                  2. complex
                    1. defining properties
                      1. closed under vector addition and scalar multiplication (linear combination)
                        1. Interaction
                          1. Vector Algebra
                            1. Vector Multiplication
                              1. Scaling
                                1. Two Vectors
                                  1. Dot Product
                                    1. Einstein Summation Notation
                                      1. Algebraic Notation
                                        1. Intuition
                                          1. Vector Projection; Observe "overlap degree" of two vector quantities
                                            1. small-scale operation (vector-on-vector)
                                          2. Cross Product
                                            1. Levi-Civita Notation
                                              1. Levi-Civita (Permutation) Symbol
                                                1. E123=1
                                                  1. Antisymmetric (swap sign per each index swap)
                                                2. Algebraic Notation
                                                  1. Intuition
                                                    1. Defines Perpendicular Vector ("area of plane" vector)
                                                3. Three Vectors
                                                  1. Triple Scalar Product (cyclically permutable)
                                                    1. Defines 3-D Volume (3-D Determinant)
                                                    2. Triple Vector Product (obeys cyclic permutation)
                                                  2. Vector Addition
                                                    1. Component by Component
                                            2. mathematically represent models of multi-element physical systems
                                            3. matrices (nxn or nxm)
                                        2. Topological Spaces
                                          1. Hilbert Spaces
                                            1. others
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