Multivariable Calculus

  1. Vector-Valued Functions and Space Curves
    1. Introduction to Vector-Valued Functions
      1. Definition and Notation
        1. Domain and Range
          1. Examples of Vector Functions
            1. Parametric Representation of Curves
            2. Limits and Continuity of Vector Functions
              1. Limits of Vector Functions
                1. Component-wise Limits
                  1. Properties of Limits
                  2. Continuity of Vector Functions
                    1. Graphing Space Curves
                    2. Derivatives of Vector Functions
                      1. Definition of the Derivative
                        1. Component-wise Differentiation
                          1. Geometric Interpretation
                            1. Tangent Vectors
                              1. Velocity Vectors
                              2. Differentiation Rules
                                1. Sum and Difference Rules
                                  1. Scalar Multiple Rule
                                    1. Product Rules
                                      1. Dot Product Rule
                                        1. Cross Product Rule
                                        2. Chain Rule for Vector Functions
                                      2. Integration of Vector Functions
                                        1. Indefinite Integrals
                                          1. Component-wise Integration
                                            1. Antiderivatives of Vector Functions
                                            2. Definite Integrals
                                              1. Fundamental Theorem for Vector Functions
                                                1. Displacement and Net Change
                                              2. Arc Length and Parametrization
                                                1. Arc Length of Space Curves
                                                  1. Arc Length Formula
                                                    1. Computing Arc Length
                                                    2. Arc Length Function
                                                      1. Parametrization by Arc Length
                                                        1. Reparametrization of Curves
                                                        2. Curvature and the TNB Frame
                                                          1. Curvature
                                                            1. Definition and Formula
                                                              1. Curvature for Plane Curves
                                                                1. Curvature for Space Curves
                                                                  1. Radius of Curvature
                                                                  2. Unit Tangent Vector
                                                                    1. Principal Unit Normal Vector
                                                                      1. Binormal Vector
                                                                        1. The Frenet Frame
                                                                          1. Construction and Properties
                                                                            1. Geometric Interpretation
                                                                            2. Torsion
                                                                              1. Definition and Computation
                                                                                1. Physical Meaning
                                                                              2. Motion in Space
                                                                                1. Position, Velocity, and Acceleration
                                                                                  1. Position Vector Functions
                                                                                    1. Velocity as Derivative of Position
                                                                                      1. Acceleration as Derivative of Velocity
                                                                                        1. Speed vs Velocity
                                                                                        2. Components of Acceleration
                                                                                          1. Tangential Component
                                                                                            1. Normal Component
                                                                                              1. Physical Interpretation
                                                                                              2. Applications to Physics
                                                                                                1. Projectile Motion
                                                                                                  1. Circular and Orbital Motion
                                                                                                    1. Planetary Motion and Kepler's Laws

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                                                                                                1. Foundations of Three-Dimensional Space

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                                                                                                3. Functions of Several Variables