Tuesday, August 16, 2016

Calculus and Higher Calculus

Splitting Calculus and Analysis into their components. Some components are underrated for the amount of time necessary for them. Univariate Calculus and Multivariate Calculus are not underrated. Typically Univariate Calculus forms two 4 unit classes. Multivariate Calculus would join with Vectors to form a 4 unit class.
  • Univariate Differential Calculus (3 Units)
    • Univariate Integral Calculus (3 Units)
      • Univariate Series Calculus  (2 Units)
        • Limit Analysis (2 Units)
          • Real Differential Analysis (2 Units)
            • Real Integral Analysis (2 Units)
              • Real Sequence and Series Analysis (2 Units)
            • Real Multivariate Differential Analysis (2 Units) (Additional Prerequisite: Multivariate Calculus)
              • Real Multivariate Integral Analysis (2 Units)
                • Real Multivariate Sequence and Series Analysis (2 Units)
    • Multivariate Differential Calculus (1 Unit) (Additional Prerequisite: Vectors (1 Unit))
      • Multivariate Integral Calculus (1 Unit) (Additional Prerequisite: Univariate Integral Calculus)
        • Multivariate Series Calculus (1 Unit) (Additional Prerequisite: Univariate Series Calculus)
        • Vector Differential Calculus (2 Units)
          • Vector Integral Calculus (2 Units)
            • Vector Differential Analysis (2 Units) (Additional Prerequisite: Limit Analysis) 
              • Vector Integral Analysis (2 Units)
        • Complex Differential Calculus (1 Unit) (Additional Prerequisite: Univariate Series Calculus)
          •  Complex Integral Calculus (1 Unit)
            • Complex Series Calculus (1 Unit)
            • Complex Differential Analysis (1 Unit) (Additional Prerequisite: Real Analysis)
              •  Complex Integral Analysis (1 Unit)
                • Complex Series Analysis (1 Unit) (Additional Prerequisite: Complex Series Calculus)
                • Multivariate Complex Differential Analysis (2 Units)
                  • Multivariate Complex Integral Analysis (2 Units)
                    • Multivariate Complex Series Analysis (2 Units) (Additional Prerequisite: Complex Series Analysis)
        • Fourier Series (1 Unit) (Additional Requirement: Univariate Series Calculus)
          • Fourier Analysis (1 Unit)
            • Multivariate Fourier Analysis (1 Unit)
Limit Analysis = Introduction to Real Analysis

Limit Analysis is using sequences, series, and epsilon to rigorously define the limit. Infimum, Supremum, Limit Inferior, Limit Superior are other key component that needs to be rigorously defined with epsilonics. Limit Superior is necessary for a proof to prove the Chain Rule. There are more complicated alternatives.


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