1.1.1. | The Domain of a Vector-Valued Function | |
1.1.2. | Limits of Vector-Valued Functions | |
1.1.3. | Continuity and Differentiability of Vector-Valued Functions | |
1.1.4. | Differentiation Rules for Vector-Valued Functions | |
1.1.5. | Integration Rules for Vector-Valued Functions | |
1.1.6. | The Arc Length of a Vector-Valued Function | |
1.1.7. | Tangent Vectors and Tangent Lines to Curves | |
1.1.8. | Unit Tangent Vectors | |
1.1.9. | Principal Normal Vectors | |
1.1.10. | Binormal Vectors | |
1.1.11. | The Osculating Plane | |
1.1.12. | Parameterization by Arc Length |
1.2.1. | Introduction to Curvature | |
1.2.2. | Finding Curvature Using the Cross Product | |
1.2.3. | Radius of Curvature | |
1.2.4. | The Curvature of a Plane Curve |
1.3.1. | Vector Fields | |
1.3.2. | Visualizing Vector Fields | |
1.3.3. | Gradient Vector Fields | |
1.3.4. | Conservative Vector Fields in the Cartesian Plane | |
1.3.5. | Calculating Potential Functions | |
1.3.6. | Connected and Simply-Connected Regions | |
1.3.7. | Stream Functions |
1.4.1. | The Divergence of a Vector Field | |
1.4.2. | Properties of the Divergence Operator | |
1.4.3. | The Curl of a Vector Field | |
1.4.4. | Properties of the Curl Operator |
2.5.1. | Interior and Boundary Points | |
2.5.2. | Interiors and Boundaries of Sets | |
2.5.3. | Open and Closed Sets | |
2.5.4. | The Domain of a Multivariable Function | |
2.5.5. | Level Curves | |
2.5.6. | Level Surfaces | |
2.5.7. | Limits and Continuity of Multivariable Functions |
2.6.1. | Ellipsoids | |
2.6.2. | Hyperboloids | |
2.6.3. | Paraboloids | |
2.6.4. | Elliptic Cones | |
2.6.5. | Cylinders | |
2.6.6. | Intersections of Lines and Planes With Surfaces | |
2.6.7. | Identifying Quadric Surfaces |
2.7.1. | Introduction to Partial Derivatives | |
2.7.2. | Computing Partial Derivatives Using the Rules of Differentiation | |
2.7.3. | Geometric Interpretations of Partial Derivatives | |
2.7.4. | Partial Differentiability of Multivariable Functions | |
2.7.5. | Higher-Order Partial Derivatives | |
2.7.6. | Equality of Mixed Partial Derivatives | |
2.7.7. | Tangent Planes to Surfaces | |
2.7.8. | Linearization of Multivariable Functions |
2.8.1. | The Gradient Vector | |
2.8.2. | The Gradient as a Normal Vector | |
2.8.3. | Tangent Lines to Level Curves | |
2.8.4. | Tangent Planes to Level Surfaces | |
2.8.5. | Directional Derivatives | |
2.8.6. | The Multivariable Mean-Value Theorem |
2.9.1. | The Multivariable Chain Rule | |
2.9.2. | The Multivariable Chain Rule With Polar Coordinates | |
2.9.3. | The Multivariable Chain Rule in Vector Form | |
2.9.4. | Differentials |
2.10.1. | Affine Transformations | |
2.10.2. | The Image of an Affine Transformation | |
2.10.3. | The Inverse of an Affine Transformation | |
2.10.4. | Nonlinear Transformations of Plane Regions | |
2.10.5. | Polar Coordinate Transformations | |
2.10.6. | Transformations of Regions Between Curves |
2.11.1. | The Jacobian | |
2.11.2. | The Inverse Function Theorem | |
2.11.3. | The Jacobian of a Three-Dimensional Transformation | |
2.11.4. | The Derivative of a Multivariable Function | |
2.11.5. | The Second Derivative of a Multivariable Function | |
2.11.6. | Second-Degree Taylor Polynomials of Multivariable Functions |
2.12.1. | Global vs. Local Extrema and Critical Points of Multivariable Functions | |
2.12.2. | The Second Partial Derivatives Test | |
2.12.3. | Calculating Global Extrema of Multivariable Functions | |
2.12.4. | Lagrange Multipliers With One Constraint | |
2.12.5. | Lagrange Multipliers With Multiple Constraints | |
2.12.6. | Optimizing Multivariable Functions Using Lagrange Multipliers |
3.13.1. | Double Summations | |
3.13.2. | Partitions of Intervals | |
3.13.3. | Calculating Double Summations Over Partitions | |
3.13.4. | Approximating Volumes Using Lower Riemann Sums | |
3.13.5. | Approximating Volumes Using Upper Riemann Sums | |
3.13.6. | Lower Riemann Sums Over General Rectangular Partitions | |
3.13.7. | Upper Riemann Sums Over General Rectangular Partitions | |
3.13.8. | Defining Double Integrals Using Lower and Upper Riemann Sums |
3.14.1. | Double Integrals Over Rectangular Domains | |
3.14.2. | Double Integrals Over Non-Rectangular Domains | |
3.14.3. | Properties of Double Integrals | |
3.14.4. | Type I and II Regions in Two-Dimensional Space | |
3.14.5. | Double Integrals Over Type I Regions | |
3.14.6. | Double Integrals Over Type II Regions | |
3.14.7. | Double Integrals Over Partitioned Regions | |
3.14.8. | Changing the Order of Integration in Double Integrals |
3.15.1. | Repeated Integrals in Three Dimensions | |
3.15.2. | Triple Integrals Over Rectangular Domains | |
3.15.3. | Type I, II, and III Regions in Three-Dimensional Space | |
3.15.4. | Triple Integrals Over Type I Regions | |
3.15.5. | Triple Integrals Over Type II Regions | |
3.15.6. | Triple Integrals Over Type III Regions | |
3.15.7. | Calculating Volumes of Solids Using Triple Integrals | |
3.15.8. | Changing the Order of Integration in Triple Integrals: Changing Projection | |
3.15.9. | Changing the Order of Integration in Triple Integrals: Changing Region |
3.16.1. | Double Integrals in Plane Polar Coordinates | |
3.16.2. | Double Integrals Between Polar Curves | |
3.16.3. | Computing Areas Using a Change of Variables | |
3.16.4. | Computing Double Integrals Using a Change of Variables |
3.17.1. | Cylindrical Polar Coordinates | |
3.17.2. | Surfaces in Cylindrical Polar Coordinates | |
3.17.3. | Spherical Polar Coordinates | |
3.17.4. | Surfaces in Spherical Polar Coordinates | |
3.17.5. | Triple Integrals in Cylindrical Polar Coordinates | |
3.17.6. | Triple Integrals in Spherical Polar Coordinates | |
3.17.7. | Computing Triple Integrals Using a Change of Variables |
4.18.1. | Simple, Closed, and Oriented Curves | |
4.18.2. | Line Integrals of Scalar Functions | |
4.18.3. | Properties of Line Integrals of Scalar Functions | |
4.18.4. | Line Integrals of Scalar Functions Over Paths Expressed as Functions of X | |
4.18.5. | Line Integrals of Scalar Functions Over Paths Expressed as Functions of Y |
4.19.1. | Line Integrals of Scalar Functions Over Line Segments | |
4.19.2. | Line Integrals of Scalar Functions Over Circles | |
4.19.3. | Line Integrals of Scalar Functions Over Ellipses | |
4.19.4. | Further Properties of Line Integrals of Scalar Functions | |
4.19.5. | Line Integrals of Scalar Functions Over Polar Curves |
4.20.1. | Line Integrals With Respect to X and Y | |
4.20.2. | Properties of Line Integrals With Respect to X and Y | |
4.20.3. | Sums of Line Integrals With Respect to X and Y Over Parametric Curves | |
4.20.4. | Sums of Line Integrals With Respect to X and Y |
4.21.1. | Line Integrals of Vector-Valued Functions Over Parametric Curves | |
4.21.2. | Line Integrals of Vector-Valued Functions Over General Curves | |
4.21.3. | Interpreting Line Integrals of Vector-Valued Functions | |
4.21.4. | Properties of Line Integrals of Vector-Valued Functions | |
4.21.5. | The Fundamental Theorem for Line Integrals | |
4.21.6. | Path Independence of Line Integrals |
4.22.1. | Outward-Pointing Unit Normal Vectors in 2D | |
4.22.2. | Circulation | |
4.22.3. | Flux in Two-Dimensional Vector Fields | |
4.22.4. | Calculating Flux in Two-Dimensional Vector Fields | |
4.22.5. | Source-Free Vector Fields |
4.23.1. | Introduction to Green's Theorem | |
4.23.2. | Green's Theorem in Polar Coordinates | |
4.23.3. | Using Green's Theorem to Calculate Area | |
4.23.4. | Extending Green's Theorem | |
4.23.5. | Green's Theorem in Flux Form |
5.24.1. | The Hyperbolic Functions | |
5.24.2. | Parametric Surfaces | |
5.24.3. | Tangent Planes to Parametric Surfaces | |
5.24.4. | Parametrizations of Ellipsoids and Cones | |
5.24.5. | Parametrizations of Paraboloids and Hyperboloids | |
5.24.6. | Parametrizations of Cylinders |
5.25.1. | Surface Areas of Revolution: Rotation About the X-Axis | |
5.25.2. | Surface Areas of Revolution: Rotation About the Y-Axis | |
5.25.3. | Surface Areas of Revolution for Parametric Curves | |
5.25.4. | Areas of Parametric Surfaces | |
5.25.5. | Surfaces of Revolution | |
5.25.6. | Areas of Surfaces |
5.26.1. | Surface Integrals Over Parametric Surfaces | |
5.26.2. | Surface Integrals Over Cartesian Surfaces | |
5.26.3. | Flux in Three-Dimensional Vector Fields | |
5.26.4. | Flux Through Closed Surfaces | |
5.26.5. | Calculating Flux Through Parametric Surfaces | |
5.26.6. | Calculating Flux Through Cartesian Surfaces | |
5.26.7. | Calculating Flux Through Closed Surfaces | |
5.26.8. | The Divergence Theorem | |
5.26.9. | Stokes' Theorem |
6.27.1. | Velocity and Acceleration as Functions of Displacement | |
6.27.2. | Determining Properties of Objects Described as Functions of Displacement | |
6.27.3. | The Components of Acceleration | |
6.27.4. | Newton's Second Law | |
6.27.5. | Applying Newton's Second Law in the Plane | |
6.27.6. | The Work-Energy Principle | |
6.27.7. | Circular Motion About the Origin |
6.28.1. | The Average Value of a Multivariable Function | |
6.28.2. | Density, Mass, and Charge of Plane Laminas | |
6.28.3. | Moments and Center of Mass | |
6.28.4. | Moments and Centers of Mass of Thin Rods | |
6.28.5. | Moments and Centers of Mass of Plane Laminas | |
6.28.6. | Moments of Inertia of Laminas About the Coordinate Axes | |
6.28.7. | Moments of Inertia of Laminas About Other Axes | |
6.28.8. | Calculating the Radius of Gyration of a Plane Lamina | |
6.28.9. | The Parallel Axis Theorem |
6.29.1. | Probability Density Functions of Continuous Random Variables | |
6.29.2. | Calculating Probabilities With Continuous Random Variables | |
6.29.3. | Continuous Random Variables Over Infinite Domains | |
6.29.4. | Joint Distributions for Discrete Random Variables | |
6.29.5. | Joint Distributions for Continuous Random Variables |