Multivariable Calculus

  1. Vector Calculus
    1. Vector Fields
      1. Definition of Vector Fields
        1. Examples of Vector Fields
          1. Gravitational Fields
            1. Electric Fields
              1. Velocity Fields
              2. Visualization of Vector Fields
                1. Gradient Vector Fields
                  1. Conservative Vector Fields
                    1. Definition and Properties
                      1. Potential Functions
                        1. Test for Conservative Fields
                      2. Line Integrals
                        1. Line Integrals of Scalar Functions
                          1. Definition and Computation
                            1. Applications to Arc Length and Mass
                            2. Line Integrals of Vector Fields
                              1. Definition and Computation
                                1. Work Done by Force Fields
                                  1. Circulation
                                  2. Properties of Line Integrals
                                    1. Independence of Parametrization
                                    2. Fundamental Theorem for Line Integrals
                                      1. Statement of the Theorem
                                        1. Path Independence
                                          1. Conservative Vector Fields and Potential Functions
                                            1. Applications to Physics
                                              1. Conservative Forces
                                                1. Conservation of Energy
                                              2. Green's Theorem
                                                1. Statement of Green's Theorem
                                                  1. Conditions for Application
                                                    1. Proof Strategy
                                                      1. Applications of Green's Theorem
                                                        1. Evaluating Line Integrals
                                                          1. Computing Areas
                                                            1. Flux and Circulation
                                                            2. Vector Forms of Green's Theorem
                                                            3. Curl and Divergence
                                                              1. The Curl of a Vector Field
                                                                1. Definition in 2D and 3D
                                                                  1. Computation of Curl
                                                                    1. Physical Interpretation
                                                                      1. Curl and Conservative Fields
                                                                      2. The Divergence of a Vector Field
                                                                        1. Definition and Computation
                                                                          1. Physical Interpretation
                                                                            1. Divergence and Sources/Sinks
                                                                            2. Vector Identities
                                                                              1. Curl of Gradient
                                                                                1. Divergence of Curl
                                                                                  1. Other Important Identities
                                                                                2. Parametric Surfaces
                                                                                  1. Parametrization of Surfaces
                                                                                    1. Examples of Parametric Surfaces
                                                                                      1. Surfaces of Revolution
                                                                                        1. Graphs of Functions
                                                                                        2. Tangent Vectors and Normal Vectors
                                                                                          1. Surface Area
                                                                                            1. Surface Area Formula
                                                                                              1. Computing Surface Areas
                                                                                            2. Surface Integrals
                                                                                              1. Surface Integrals of Scalar Functions
                                                                                                1. Definition and Computation
                                                                                                  1. Applications to Mass and Center of Mass
                                                                                                  2. Surface Integrals of Vector Fields
                                                                                                    1. Flux Through Surfaces
                                                                                                      1. Definition and Computation
                                                                                                      2. Orientation of Surfaces
                                                                                                        1. Orientable Surfaces
                                                                                                          1. Choosing Normal Vectors
                                                                                                            1. Möbius Strip and Non-orientable Surfaces
                                                                                                          2. Stokes' Theorem
                                                                                                            1. Statement of Stokes' Theorem
                                                                                                              1. Relationship to Green's Theorem
                                                                                                                1. Conditions for Application
                                                                                                                  1. Physical Interpretation
                                                                                                                    1. Circulation and Curl
                                                                                                                    2. Applications of Stokes' Theorem
                                                                                                                    3. Divergence Theorem
                                                                                                                      1. Statement of the Divergence Theorem
                                                                                                                        1. Relationship Between Surface and Volume Integrals
                                                                                                                          1. Conditions for Application
                                                                                                                            1. Physical Interpretation
                                                                                                                              1. Flux and Divergence
                                                                                                                                1. Sources and Sinks
                                                                                                                                2. Applications of the Divergence Theorem
                                                                                                                                3. Unified View of Fundamental Theorems
                                                                                                                                  1. Fundamental Theorem of Calculus
                                                                                                                                    1. Fundamental Theorem for Line Integrals
                                                                                                                                      1. Green's Theorem
                                                                                                                                        1. Stokes' Theorem
                                                                                                                                          1. Divergence Theorem
                                                                                                                                            1. Common Structure and Generalizations
                                                                                                                                              1. Higher-Dimensional Analogues

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                                                                                                                                            6. Multiple Integrals

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                                                                                                                                            1. Foundations of Three-Dimensional Space