Calculus

  1. Introduction to Multivariable Calculus
    1. Three-Dimensional Coordinate Systems
      1. The Three-Dimensional Coordinate System
        1. Cartesian Coordinates in Space
          1. Distance Formula in Three Dimensions
            1. Equation of a Sphere
            2. Vectors in Space
              1. Vector Notation and Components
                1. Vector Operations
                  1. Addition and Subtraction
                    1. Scalar Multiplication
                    2. Magnitude and Unit Vectors
                    3. The Dot Product
                      1. Definition and Properties
                        1. Geometric Interpretation
                          1. Angle Between Vectors
                            1. Vector Projections
                            2. The Cross Product
                              1. Definition and Properties
                                1. Geometric Interpretation
                                  1. Applications to Area and Volume
                                  2. Lines and Planes in Space
                                    1. Vector Equations of Lines
                                      1. Parametric Equations of Lines
                                        1. Equations of Planes
                                          1. Distances and Intersections
                                        2. Vector-Valued Functions
                                          1. Vector Functions and Space Curves
                                            1. Definition and Examples
                                              1. Limits and Continuity
                                              2. Derivatives of Vector Functions
                                                1. Definition of Derivative
                                                  1. Rules for Differentiation
                                                    1. Tangent Vectors
                                                    2. Integrals of Vector Functions
                                                      1. Indefinite and Definite Integrals
                                                        1. Applications to Motion
                                                        2. Arc Length and Curvature
                                                          1. Arc Length Formula for Space Curves
                                                            1. Curvature and Its Geometric Meaning
                                                          2. Functions of Several Variables
                                                            1. Functions of Two Variables
                                                              1. Definition and Domain
                                                                1. Graphs and Level Curves
                                                                  1. Functions of Three or More Variables
                                                                  2. Limits and Continuity
                                                                    1. Limits of Functions of Several Variables
                                                                      1. Continuity in Several Variables
                                                                      2. Partial Derivatives
                                                                        1. Definition of Partial Derivatives
                                                                          1. Geometric Interpretation
                                                                            1. Higher-Order Partial Derivatives
                                                                              1. Mixed Partial Derivatives
                                                                              2. Tangent Planes and Linear Approximation
                                                                                1. Equation of Tangent Plane
                                                                                  1. Linear Approximation in Several Variables
                                                                                    1. Differentials
                                                                                    2. The Chain Rule
                                                                                      1. Chain Rule for Several Variables
                                                                                      2. Directional Derivatives and Gradients
                                                                                        1. Definition of Directional Derivative
                                                                                          1. The Gradient Vector
                                                                                            1. Properties and Applications of Gradients
                                                                                            2. Optimization in Several Variables
                                                                                              1. Critical Points
                                                                                                1. Second Derivative Test
                                                                                                  1. Constrained Optimization and Lagrange Multipliers
                                                                                                2. Multiple Integrals
                                                                                                  1. Double Integrals
                                                                                                    1. Definition Over Rectangles
                                                                                                      1. Iterated Integrals
                                                                                                        1. Fubini's Theorem
                                                                                                        2. Double Integrals Over General Regions
                                                                                                          1. Type I and Type II Regions
                                                                                                            1. Changing Order of Integration
                                                                                                              1. Applications to Area and Volume
                                                                                                              2. Double Integrals in Polar Coordinates
                                                                                                                1. Coordinate Transformation
                                                                                                                  1. Jacobian for Polar Coordinates
                                                                                                                  2. Triple Integrals
                                                                                                                    1. Definition and Evaluation
                                                                                                                      1. Applications to Volume and Mass
                                                                                                                      2. Triple Integrals in Cylindrical Coordinates
                                                                                                                        1. Cylindrical Coordinate System
                                                                                                                          1. Setting Up Triple Integrals
                                                                                                                          2. Triple Integrals in Spherical Coordinates
                                                                                                                            1. Spherical Coordinate System
                                                                                                                              1. Setting Up Triple Integrals
                                                                                                                              2. Change of Variables in Multiple Integrals
                                                                                                                                1. General Transformation
                                                                                                                                  1. Jacobian Determinant

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                                                                                                                              9. Infinite Sequences and Series

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                                                                                                                              1. Foundations for Calculus