Mathematics for Machine Learning and Data Science

  1. Multivariable Calculus: The Tools for Optimization
    1. Functions of Multiple Variables
      1. Definition and Notation
        1. Domain and Range in Higher Dimensions
          1. Level Sets and Contour Lines
            1. Visualizing Multivariable Functions
              1. Contour Plots
                1. Surface Plots
                  1. Heat Maps
                  2. Limits and Continuity
                    1. Definition of Limit in Higher Dimensions
                      1. Continuity Criteria
                        1. Properties of Continuous Functions
                      2. Partial Derivatives
                        1. Definition and Geometric Interpretation
                          1. Notation for Partial Derivatives
                            1. Computing Partial Derivatives
                              1. Higher-Order Partial Derivatives
                                1. Mixed Partial Derivatives
                                  1. Clairaut's Theorem
                                    1. Partial Derivative Rules
                                    2. The Gradient
                                      1. Definition and Notation
                                        1. Geometric Interpretation
                                          1. Properties of the Gradient
                                            1. Gradient as Normal Vector
                                              1. Level Sets and Gradients
                                              2. Directional Derivatives
                                                1. Definition and Calculation
                                                  1. Relationship to the Gradient
                                                    1. Maximum Rate of Change
                                                    2. The Chain Rule for Multivariable Functions
                                                      1. Chain Rule for Composite Functions
                                                        1. Matrix Form of the Chain Rule
                                                          1. Applications to Implicit Differentiation
                                                          2. The Jacobian Matrix
                                                            1. Definition for Vector-Valued Functions
                                                              1. Computing the Jacobian
                                                                1. Jacobian Determinant
                                                                  1. Change of Variables
                                                                    1. Applications in Transformations
                                                                    2. The Hessian Matrix
                                                                      1. Definition and Computation
                                                                        1. Interpretation as Curvature
                                                                          1. Symmetry of the Hessian
                                                                            1. Role in Optimization
                                                                              1. Positive and Negative Definiteness
                                                                              2. Taylor Series for Multivariable Functions
                                                                                1. First-Order Taylor Approximation
                                                                                  1. Second-Order Taylor Approximation
                                                                                    1. Multivariate Taylor Theorem
                                                                                      1. Applications in Optimization
                                                                                      2. Optimization of Multivariable Functions
                                                                                        1. Critical Points
                                                                                          1. Definition and Finding Critical Points
                                                                                            1. Types of Critical Points
                                                                                            2. Local Extrema
                                                                                              1. Local Maxima and Minima
                                                                                                1. Saddle Points
                                                                                                2. Second Derivative Test
                                                                                                  1. Using the Hessian Matrix
                                                                                                    1. Classification of Critical Points
                                                                                                    2. Global Extrema
                                                                                                      1. Extreme Value Theorem
                                                                                                        1. Finding Global Extrema
                                                                                                        2. Convexity and Concavity
                                                                                                          1. Convex Sets
                                                                                                            1. Convex Functions
                                                                                                              1. Properties of Convex Functions
                                                                                                                1. Implications for Optimization
                                                                                                                2. Constrained Optimization
                                                                                                                  1. Equality Constraints
                                                                                                                    1. Lagrange Multipliers
                                                                                                                      1. Inequality Constraints
                                                                                                                        1. Karush-Kuhn-Tucker Conditions
                                                                                                                      2. Multiple Integrals
                                                                                                                        1. Double Integrals
                                                                                                                          1. Definition and Computation
                                                                                                                            1. Iterated Integrals
                                                                                                                              1. Change of Order of Integration
                                                                                                                              2. Triple Integrals
                                                                                                                                1. Change of Variables in Multiple Integrals
                                                                                                                                  1. Applications to Probability

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                                                                                                                                4. Probability Theory: Modeling Uncertainty