81 lines
2.4 KiB
Markdown
81 lines
2.4 KiB
Markdown
# Mathematics
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This is my attempt at digitalizing (and translating) my math notes from uni.
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It contains all the maths I learned in four semesters of university, however I have not proofread it yet, and I also plan on adding in all the proofs that were labeled as "left to the lecture attendant".
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The topics covered in this script will be:
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1. Fundamentals and Notation
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1.1 Logic
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1.2 Sets and Functions
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1.3 Numbers
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2. Analysis: Part 1
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2.1 Elementary Inequalities
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2.2 Sequences and Limits
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2.3 Convergence of Series
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3. Linear Algebra
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3.1 Vector spaces
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3.2 Matrices and Gaussian elimination
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3.3 The Determinant
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3.4 Scalar Product
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3.5 Eigenvalue problems
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4. Analysis: Part 2
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4.1 Limits of Functions
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4.2 Differential Calculus
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5. Topology in Metric spaces
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5.1 Metric and Normed spaces
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5.2 Sequences, Series and Limits
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5.3 Open and Closed Sets
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5.4 Continuiuty
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5.5 Convergence of Function Sequences
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6. Differential Calculus for Functions with multiple Variables
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6.1 Partial and Total Differentiability
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6.2 Higher Derivatives
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6.3 Function Sequences and Differentiability
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6.4 The Banach Fixed-Point Theorem and the Implicit Function Theorem
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7. Measures and Integrals
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7.1 Contents and Measures
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7.2 Integrals
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7.3 Integrals over the real numbers
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7.4 Product Measures and the Fubini Theorem
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7.5 The Transformation Theorem
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7.6 Lebesgue-Stieltjes Integral
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8. Ordinary Differential Equations
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8.1 Solution Methods
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8.2 The Picard-Lindelöf Theorem
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8.3 Linear Differential Equation Systems
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9. Integration over Submanifolds
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9.1 Line Integrals
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9.2 Surface Integrals
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9.3 Ingegral Theorems
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10. Elements of Complex Analysis
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10.1 Complex Differentiability
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10.2 Complex Line Integrals
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10.3 Identity Theorems and Analytic Continuation
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10.4 Laurent Series
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10.5 Residue Theorem
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10.6 Application: Potential Theory
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11. Fourier Transform and Basics of Distribution Theory
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11.1 Fourier Transform on L¹(ℝᵈ)
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11.2 Fourier Transform on L²(ℝᵈ)
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11.3 Outlook: Tempered Distributions
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12. Operator Theory
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12.1 Linear Operators
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12.2 Dual Spaces
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12.3 Hilbert Spaces
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12.4 Adjoint Operators
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13. Spectral Theory
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13.1 Spectral Theory of Bounded Linear Operators
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13.2 Spectral Representation of Bounded Self-Adjoint Operators
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13.3 Compact & Unbounded Linear Operators |