1.1.1. | Interior and Boundary Points | |
1.1.2. | Interiors and Boundaries of Sets | |
1.1.3. | Open and Closed Sets |
1.2.1. | Limits Involving the Exponential Function |
1.3.1. | Parametric Equations of Circles | |
1.3.2. | Parametric Equations of Ellipses | |
1.3.3. | Parametric Equations of Parabolas | |
1.3.4. | Parametric Equations of Parabolas Centered at (h,k) |
2.4.1. | Spheres as Quadric Surfaces | |
2.4.2. | Ellipsoids as Quadric Surfaces | |
2.4.3. | Hyperboloids as Quadric Surfaces | |
2.4.4. | Paraboloids as Quadric Surfaces | |
2.4.5. | Elliptic Cones as Quadric Surfaces | |
2.4.6. | Cylinders as Quadric Surfaces | |
2.4.7. | Identifying Quadric Surfaces |
2.5.1. | Simple, Closed, and Oriented Curves |
3.6.1. | Defining Vector-Valued Functions | |
3.6.2. | Limits of Vector-Valued Functions | |
3.6.3. | Continuity and Differentiability of Vector-Valued Functions | |
3.6.4. | Differentiation Rules for Vector-Valued Functions | |
3.6.5. | Integrals of Vector-Valued Functions | |
3.6.6. | Properties of Integrals of Vector-Valued Functions |
3.7.1. | Tangent Vectors and Tangent Lines to Curves | |
3.7.2. | Unit Tangent Vectors | |
3.7.3. | Principal Normal Vectors | |
3.7.4. | Binormal Vectors | |
3.7.5. | The Osculating Plane | |
3.7.6. | The Arc Length of a Vector-Valued Function | |
3.7.7. | Parameterization of a Curve by Arc Length |
3.8.1. | Introduction to Curvature | |
3.8.2. | Finding Curvature Using the Cross Product | |
3.8.3. | Radius of Curvature | |
3.8.4. | The Curvature of a Plane Curve | |
3.8.5. | Intrinsic Coordinates |
4.9.1. | Introduction to Multivariable Functions | |
4.9.2. | Level Curves and Contour Plots | |
4.9.3. | Level Surfaces of Multivariable Functions | |
4.9.4. | Limits and Continuity of Multivariable Functions |
4.10.1. | Introduction to Partial Derivatives | |
4.10.2. | Computing Partial Derivatives Using the Rules of Differentiation | |
4.10.3. | Geometric Interpretations of Partial Derivatives | |
4.10.4. | Partial Differentiability of Multivariable Functions | |
4.10.5. | Higher-Order Partial Derivatives | |
4.10.6. | Equality of Mixed Partial Derivatives |
4.11.1. | Tangent Planes to Surfaces | |
4.11.2. | Linearization of Multivariable Functions | |
4.11.3. | Further Differentiability of Multivariable Functions |
4.12.1. | The Gradient Vector | |
4.12.2. | The Gradient as a Normal Vector | |
4.12.3. | Tangent Lines to Level Curves | |
4.12.4. | Tangent Planes to Level Surfaces | |
4.12.5. | Directional Derivatives | |
4.12.6. | The Multivariable Mean-Value Theorem |
4.13.1. | The Multivariable Chain Rule | |
4.13.2. | The Multivariable Chain Rule With Polar Coordinates | |
4.13.3. | The Multivariable Chain Rule in Vector Form | |
4.13.4. | Implicit Differentiation of Multivariable Functions | |
4.13.5. | Differentials |
4.14.1. | Defining Local and Global Extrema of Multivariable Functions | |
4.14.2. | Critical Points of Multivariable Functions | |
4.14.3. | The Second Partial Derivatives Test | |
4.14.4. | Global Extrema of Multivariable Functions | |
4.14.5. | The Hessian Matrix | |
4.14.6. | Constrained Optimization | |
4.14.7. | The Method of Lagrange Multipliers With One Constraint | |
4.14.8. | The Method of Lagrange Multipliers With Multiple Constraints |
5.15.1. | Double Summations | |
5.15.2. | Partitions of Intervals | |
5.15.3. | Calculating Double Summations Over Partitions | |
5.15.4. | Approximating Volumes Using Lower Riemann Sums | |
5.15.5. | Approximating Volumes Using Upper Riemann Sums | |
5.15.6. | Lower Riemann Sums Over General Rectangular Partitions | |
5.15.7. | Upper Riemann Sums Over General Rectangular Partitions | |
5.15.8. | Defining Double Integrals Using Lower and Upper Riemann Sums |
5.16.1. | Double Integrals Over Rectangular Domains | |
5.16.2. | Double Integrals Over Non-Rectangular Domains | |
5.16.3. | Properties of Double Integrals |
5.17.1. | Type I and II Regions in Two-Dimensional Space | |
5.17.2. | Double Integrals Over Type I Regions | |
5.17.3. | Double Integrals Over Type II Regions | |
5.17.4. | Double Integrals Over Partitioned Regions | |
5.17.5. | Changing the Order of Integration in Double Integrals |
5.18.1. | Type I, II, and III Regions in Three-Dimensional Space | |
5.18.2. | Repeated Integrals in Three Dimensions | |
5.18.3. | Triple Integrals Over Rectangular Domains | |
5.18.4. | Triple Integrals Over Type I Regions | |
5.18.5. | Triple Integrals Over Type II Regions | |
5.18.6. | Triple Integrals Over Type III Regions | |
5.18.7. | Calculating Volumes of Solids Using Triple Integrals | |
5.18.8. | Computing Triple Integrals Using Coordinate Plane Projections | |
5.18.9. | Changing the Order of Integration in Triple Integrals: Changing Projection | |
5.18.10. | Changing the Order of Integration in Triple Integrals: Changing Region | |
5.18.11. | Triple Integrals Over Partitioned Regions |
5.19.1. | Calculating the Mass of a Plane Laminar | |
5.19.2. | Calculating the Electric Charge of a Plane Laminar | |
5.19.3. | Calculating the Center of Mass of a Plane Laminar | |
5.19.4. | The Moment of Inertia of a Plane Laminar | |
5.19.5. | The Radius of Gyration of a Plane Laminar | |
5.19.6. | The Parallel Axis Theorem Applied to Plane Laminars | |
5.19.7. | The Average Value of a Function Over a Plane Region |
6.20.1. | Affine Transformations | |
6.20.2. | The Image of an Affine Transformation | |
6.20.3. | The Inverse of an Affine Transformation | |
6.20.4. | The Jacobian Determinant | |
6.20.5. | The Inverse Function Theorem | |
6.20.6. | Nonlinear Transformations of Plane Regions | |
6.20.7. | Polar Coordinate Transformations | |
6.20.8. | Transformations of Regions Between Curves |
6.21.1. | Double Integrals in Plane Polar Coordinates | |
6.21.2. | Double Integrals Between Polar Curves | |
6.21.3. | Computing Areas Using a Change of Variables | |
6.21.4. | Computing Double Integrals Using a Change of Variables | |
6.21.5. | Computing Improper Double Integrals Using a Change of Variables |
6.22.1. | Cylindrical Polar Coordinates | |
6.22.2. | Surfaces in Cylindrical Polar Coordinates | |
6.22.3. | Spherical Polar Coordinates | |
6.22.4. | Surfaces in Spherical Polar Coordinates |
6.23.1. | The Jacobian of a Three-Dimensional Transformation | |
6.23.2. | Computing Triple Integrals Using a Change of Variables | |
6.23.3. | Triple Integrals in Cylindrical Polar Coordinates | |
6.23.4. | Triple Integrals in Spherical Polar Coordinates |
7.24.1. | Vector Fields | |
7.24.2. | Visualizing Vector Fields | |
7.24.3. | Gradient Vector Fields | |
7.24.4. | Conservative Vector Fields in the Cartesian Plane | |
7.24.5. | Calculating Potential Functions | |
7.24.6. | Connected and Simply-Connected Regions | |
7.24.7. | Conservative Vector Fields Over Simply Connected Regions |
7.25.1. | The Divergence of a Vector Field | |
7.25.2. | Properties of the Divergence Operator | |
7.25.3. | The Curl of a Vector Field | |
7.25.4. | Properties of the Curl Operator | |
7.25.5. | Properties of the Del Operator |
8.26.1. | Line Integrals of Scalar Functions | |
8.26.2. | Properties of Line Integrals of Scalar Functions | |
8.26.3. | Line Integrals of Scalar Functions Over Paths Expressed as Functions of X | |
8.26.4. | Line Integrals of Scalar Functions Over Paths Expressed as Functions of Y |
8.27.1. | Line Integrals of Scalar Functions Over Line Segments | |
8.27.2. | Line Integrals of Scalar Functions Over Circles | |
8.27.3. | Line Integrals of Scalar Functions Over Ellipses | |
8.27.4. | Further Properties of Line Integrals of Scalar Functions | |
8.27.5. | Line Integrals of Scalar Functions Over Polar Curves |
8.28.1. | Line Integrals With Respect to X and Y | |
8.28.2. | Properties of Line Integrals With Respect to X and Y | |
8.28.3. | Sums of Line Integrals With Respect to X and Y Over Parametric Curves | |
8.28.4. | Sums of Line Integrals With Respect to X and Y |
8.29.1. | Line Integrals of Vector-Valued Functions Over Parametric Curves | |
8.29.2. | Line Integrals of Vector-Valued Functions Over General Curves | |
8.29.3. | Interpreting Line Integrals of Vector-Valued Functions | |
8.29.4. | Properties of Line Integrals of Vector-Valued Functions | |
8.29.5. | The Fundamental Theorem for Line Integrals | |
8.29.6. | Path Independence of Line Integrals | |
8.29.7. | Conservation of Energy |
8.30.1. | Outward-Pointing Unit Normal Vectors in 2D | |
8.30.2. | Circulation | |
8.30.3. | Flux in Two-Dimensional Vector Fields | |
8.30.4. | Calculating Flux in Two-Dimensional Vector Fields |
8.31.1. | Green's Theorem | |
8.31.2. | Green's Theorem in Polar Coordinates | |
8.31.3. | Using Green's Theorem to Calculate Area | |
8.31.4. | Extending Green's Theorem | |
8.31.5. | Greens Theorem Applied to Regions Containing Singularities | |
8.31.6. | Green's Theorem in Flux Form | |
8.31.7. | Stream Functions | |
8.31.8. | Source-Free Vector Fields |
8.32.1. | Calculating the Mass of a Wire | |
8.32.2. | Calculating the Electric Charge of a Wire | |
8.32.3. | Calculating the Center of Mass of a Wire | |
8.32.4. | The Moment of Inertia of a Wire | |
8.32.5. | The Radius of Gyration of a Wire | |
8.32.6. | The Average Value of a Function Over a Curve | |
8.32.7. | Calculating the Work Done by a Force Along a Curve |
9.33.1. | Parametric Surfaces | |
9.33.2. | Parametrizations of Spheres, Ellipsoids and Cones | |
9.33.3. | Parametrizations of Cylinders | |
9.33.4. | Parametrizations of Hyperboloids and Paraboloids | |
9.33.5. | Tangent Planes to Parametric Surfaces |
9.34.1. | Surface Areas of Revolution: Rotation About the X-Axis | |
9.34.2. | Surface Areas of Revolution: Rotation About the Y-Axis | |
9.34.3. | Surface Areas of Revolution for Parametric Curves | |
9.34.4. | Areas of Parametric Surfaces | |
9.34.5. | Surfaces of Revolution | |
9.34.6. | Areas of Surfaces |
9.35.1. | Surface Integrals Over Parametric Surfaces | |
9.35.2. | Surface Integrals Over Cartesian Surfaces | |
9.35.3. | Flux in Three-Dimensional Vector Fields | |
9.35.4. | Flux Through Closed Surfaces | |
9.35.5. | Calculating Flux Through Parametric Surfaces | |
9.35.6. | Calculating Flux Through Cartesian Surfaces | |
9.35.7. | Calculating Flux Through Closed Surfaces | |
9.35.8. | Surface Integrals in Cylindrical Polar Coordinates | |
9.35.9. | Surface Integrals in Spherical Polar Coordinates |
9.36.1. | The Divergence Theorem | |
9.36.2. | The Divergence Theorem With Composite Surfaces | |
9.36.3. | Stokes' Theorem | |
9.36.4. | Ampere’s Law | |
9.36.5. | Faraday’s Law of Electromagnetic Induction |