1.1.1. | Special Sets | |
1.1.2. | Set-Builder Notation | |
1.1.3. | Equivalent Sets | |
1.1.4. | Cardinality of Sets | |
1.1.5. | Subsets | |
1.1.6. | The Complement of a Set | |
1.1.7. | The Difference of Sets | |
1.1.8. | The Cartesian Product |
1.2.1. | Finite Linear Series | |
1.2.2. | Finite Quadratic Series | |
1.2.3. | Finite Cubic Series |
1.3.1. | Special Limits Involving the Exponential Function | |
1.3.2. | Further Special Limits Involving the Exponential Function |
1.4.1. | Affine Transformations | |
1.4.2. | The Image of an Affine Transformation | |
1.4.3. | The Inverse of an Affine Transformation | |
1.4.4. | Nonlinear Transformations of Plane Regions |
1.6.1. | Parametric Equations of Circles | |
1.6.2. | Parametric Equations of Ellipses | |
1.6.3. | Parametric Equations of Parabolas Centered at the Origin |
2.5.1. | The Vector Equation of a Line | |
2.5.2. | The Parametric Equations of a Line | |
2.5.3. | Writing the Cartesian Equation of a Line from the Vector Equation | |
2.5.4. | Finding the Vector Equation of a Plane Using the Dot Product | |
2.5.5. | The Cartesian Equation of a Plane | |
2.5.6. | The Parametric Equations of a Plane | |
2.5.7. | The Intersection of Two Planes | |
2.5.8. | The Shortest Distance Between a Plane and a Point |
2.7.1. | Spheres as Quadric Surfaces | |
2.7.2. | Ellipsoids as Quadric Surfaces | |
2.7.3. | Hyperboloids as Quadric Surfaces | |
2.7.4. | Paraboloids as Quadric Surfaces | |
2.7.5. | Elliptic Cones as Quadric Surfaces | |
2.7.6. | Cylinders as Quadric Surfaces | |
2.7.7. | Identifying Quadric Surfaces |
2.8.1. | Open and Closed Sets in the Plane | |
2.8.2. | Type I and Type II Regions in Two-Dimensional Space | |
2.8.3. | Simple, Closed, and Oriented Curves | |
2.8.4. | Connected and Simply-Connected Regions | |
2.8.5. | Curvilinear Coordinate Systems |
2.9.1. | Type I, Type II and Type III Regions in Three-Dimensional Space | |
2.9.2. | Cylindrical Polar Coordinates | |
2.9.3. | Surfaces in Cylindrical Polar Coordinates | |
2.9.4. | Surfaces in Cylindrical Polar Coordinates: Converting Between Cartesian and Polar Coordinates | |
2.9.5. | Spherical Polar Coordinates | |
2.9.6. | Surfaces in Spherical Polar Coordinates | |
2.9.7. | Surfaces in Spherical Polar Coordinates: Converting Between Cartesian and Polar Coordinates | |
2.9.8. | Orientable vs. Non-Orientable Surfaces |
3.10.1. | Defining Vector-Valued Functions | |
3.10.2. | Limits of Vector-Valued Functions | |
3.10.3. | Derivatives of Vector-Valued Functions | |
3.10.4. | Continuity and Differentiability of Vector-Valued Functions | |
3.10.5. | The Rules of Differentiation for Vector-Valued Functions | |
3.10.6. | Integrals of Vector-Valued Functions | |
3.10.7. | Properties of Integrals of Vector-Valued Functions |
3.11.1. | The Vector Parameterization of a Curve | |
3.11.2. | Reversing the Direction of a Curve | |
3.11.3. | Tangent Vectors and Tangent Lines to Curves | |
3.11.4. | Intersecting Curves | |
3.11.5. | Unit Tangent Vectors | |
3.11.6. | Principal Normal Vectors | |
3.11.7. | Binormal Vectors | |
3.11.8. | The Osculating Plane | |
3.11.9. | The Arc Length of a Vector-Valued Function | |
3.11.10. | Parameterization of a Curve by Arc Length |
3.12.1. | Introduction to Curvature | |
3.12.2. | Finding Curvature Using the Cross Product | |
3.12.3. | Radius of Curvature | |
3.12.4. | The Curvature of a Plane Curve | |
3.12.5. | Intrinsic Coordinates |
4.13.1. | Introduction to Multivariable Functions | |
4.13.2. | Level Curves and Contour Plots | |
4.13.3. | Level Surfaces of Multivariable Functions | |
4.13.4. | Limits and Continuity of Multivariable Functions |
4.14.1. | Introduction to Partial Derivatives | |
4.14.2. | Applying the Limit Definition of a Partial Derivative | |
4.14.3. | Geometric Interpretations of Partial Derivatives | |
4.14.4. | Partial Differentiability of Multivariable Functions | |
4.14.5. | Higher-Order Partial Derivatives | |
4.14.6. | Equality of Mixed Partial Derivatives |
4.15.1. | Tangent Planes to Surfaces | |
4.15.2. | Linearization of Multivariable Functions | |
4.15.3. | Further Differentiability of Multivariable Functions |
4.16.1. | The Gradient Vector | |
4.16.2. | The Gradient as a Normal Vector | |
4.16.3. | Tangent Lines to Level Curves | |
4.16.4. | Tangent Planes to Level Surfaces | |
4.16.5. | Directional Derivatives | |
4.16.6. | The Multivariable Mean-Value Theorem |
4.17.1. | The Multivariable Chain Rule | |
4.17.2. | The Multivariable Chain Rule With Polar Coordinates | |
4.17.3. | The Multivariable Chain Rule in Vector Form | |
4.17.4. | Implicit Differentiation of Multivariable Functions | |
4.17.5. | Differentials |
4.18.1. | Defining Local and Global Extrema of Multivariable Functions | |
4.18.2. | Calculating Critical Points of Multivariable Functions | |
4.18.3. | The Second Partial Derivatives Test | |
4.18.4. | Calculating Global Extrema of Multivariable Functions | |
4.18.5. | Constrained Optimization | |
4.18.6. | Constraints and Gradients | |
4.18.7. | The Method of Lagrange Multipliers With One Constraint | |
4.18.8. | The Method of Lagrange Multipliers With Multiple Constraints |
5.19.1. | Double Summations | |
5.19.2. | Computing Double Sums Using Known Results | |
5.19.3. | Partitions of Intervals | |
5.19.4. | Calculating Double Summations Over Partitions | |
5.19.5. | Approximating Volumes Using Lower Riemann Sums | |
5.19.6. | Approximating Volumes Using Upper Riemann Sums | |
5.19.7. | Lower Riemann Sums Over General Rectangular Partitions | |
5.19.8. | Upper Riemann Sums Over General Rectangular Partitions | |
5.19.9. | Defining Double Integrals Using Lower and Upper Riemann Sums |
5.20.1. | Evaluating Double Integrals Over Rectangular Domains | |
5.20.2. | Defining Double Integrals Over Non-Rectangular Domains | |
5.20.3. | Properties of Double Integrals |
5.21.1. | Double Integrals Over Type I Regions | |
5.21.2. | Double Integrals Over Type II Regions | |
5.21.3. | Double Integrals Over Partitioned Regions | |
5.21.4. | Calculating Volumes of Solids Using Double Integrals | |
5.21.5. | Swapping the Order of Integration in Double Integrals |
5.22.1. | Evaluating Double Integrals Using a Change of Variables | |
5.22.2. | Calculating Double Integrals Using Plane Polar Coordinates | |
5.22.3. | Double Integrals of Functions Defined Between Two Polar Curves |
5.23.1. | Calculating the Mass of a Plane Laminar | |
5.23.2. | Calculating the Electric Charge of a Plane Laminar | |
5.23.3. | Calculating the Center of Mass of a Plane Laminar | |
5.23.4. | The Moment of Inertia of a Plane Laminar | |
5.23.5. | The Radius of Gyration of a Plane Laminar | |
5.23.6. | The Parallel Axis Theorem Applied to Plane Laminars | |
5.23.7. | The Average Value of a Function Over a Plane Region |
6.24.1. | Triple Summations | |
6.24.2. | Approximating Triple Integrals Using Riemann Sums | |
6.24.3. | Riemann Sums Over General Partitions of Rectangular Solids | |
6.24.4. | Evaluating Triple Integrals Over Rectangular Domains | |
6.24.5. | Defining Triple Integrals Over Non-Rectangular Domains | |
6.24.6. | Properties of Triple Integrals |
6.25.1. | Triple Integrals Over Type I Regions | |
6.25.2. | Triple Integrals Over Type II Regions | |
6.25.3. | Triple Integrals Over Type III Regions | |
6.25.4. | Triple Integrals Over Partitioned Regions | |
6.25.5. | Calculating Volumes of Solids Using Triple Integrals | |
6.25.6. | Swapping the Order of Integration in Triple Integrals |
6.26.1. | Evaluating Triple Integrals Using a Change of Variables | |
6.26.2. | Triple Integrals in Cylindrical Polar Coordinates | |
6.26.3. | Triple Integrals in Spherical Polar Coordinates |
7.27.1. | Vectors in N-Dimensional Euclidean Space | |
7.27.2. | Vector Fields | |
7.27.3. | Visualizing Vector Fields | |
7.27.4. | Gradient Vector Fields | |
7.27.5. | Conservative Vector Fields Defined Over the Cartesian Plane | |
7.27.6. | Conservative Vector Fields With Arbitrary Domains | |
7.27.7. | Calculating a Potential Function for a Conservative Vector Field |
7.28.1. | The Jacobian of Functions of Two Variables | |
7.28.2. | The Jacobian of Functions of Three Variables | |
7.28.3. | Interpreting the Jacobian | |
7.28.4. | Linearization of Vector Fields | |
7.28.5. | The Hessian Matrix |
7.29.1. | The Divergence of a Vector Field | |
7.29.2. | Properties of the Divergence Operator | |
7.29.3. | The Curl of a Vector Field | |
7.29.4. | Properties of the Curl Operator | |
7.29.5. | Interpreting Divergence | |
7.29.6. | Interpreting Curl | |
7.29.7. | Properties of the Del Operator |
8.30.1. | Line Integrals of Scalar Functions | |
8.30.2. | Properties of Line Integrals of Scalar Functions | |
8.30.3. | Line Integrals of Scalar Functions Over Paths Expressed as Functions of X | |
8.30.4. | Line Integrals of Scalar Functions Over Paths Expressed as Functions of Y |
8.31.1. | Line Integrals of Scalar Functions Over Line Segments | |
8.31.2. | Line Integrals of Scalar Functions Over Circles | |
8.31.3. | Line Integrals of Scalar Functions Over Ellipses | |
8.31.4. | Further Properties of Line Integrals of Scalar Functions | |
8.31.5. | Line Integrals of Scalar Functions Over Polar Curves |
8.32.1. | Line Integrals With Respect to X and Y | |
8.32.2. | Properties of Line Integrals With Respect to X and Y | |
8.32.3. | Sums of Line Integrals With Respect to X and Y Over Parametric Curves | |
8.32.4. | Sums of Line Integrals With Respect to X and Y |
8.33.1. | Line Integrals of Vector-Valued Functions Over Parametric Curves | |
8.33.2. | Line Integrals of Vector-Valued Functions Over General Curves | |
8.33.3. | Interpreting Line Integrals of Vector-Valued Functions | |
8.33.4. | Properties of Line Integrals of Vector-Valued Functions | |
8.33.5. | The Fundamental Theorem for Line Integrals | |
8.33.6. | Path Independence of Line Integrals | |
8.33.7. | Conservation of Energy |
8.34.1. | Outward-Pointing Unit Normal Vectors in 2D | |
8.34.2. | Circulation | |
8.34.3. | Flux in Two-Dimensional Vector Fields | |
8.34.4. | Calculating Flux in Two-Dimensional Vector Fields |
8.35.1. | Green's Theorem | |
8.35.2. | Green's Theorem in Polar Coordinates | |
8.35.3. | Using Green's Theorem to Calculate Area | |
8.35.4. | Extending Green's Theorem | |
8.35.5. | Greens Theorem Applied to Regions Containing Singularities | |
8.35.6. | Green's Theorem in Flux Form | |
8.35.7. | Stream Functions |
8.36.1. | Calculating the Mass of a Wire | |
8.36.2. | Calculating the Electric Charge of a Wire | |
8.36.3. | Calculating the Center of Mass of a Wire | |
8.36.4. | The Moment of Inertia of a Wire | |
8.36.5. | The Radius of Gyration of a Wire | |
8.36.6. | The Average Value of a Function Over a Curve | |
8.36.7. | Calculating the Work Done by a Force Along a Curve |
9.37.1. | Parametric Surfaces | |
9.37.2. | Parametrizations of Spheres, Ellipsoids and Cones | |
9.37.3. | Parametrizations of Cylinders | |
9.37.4. | Parametrizations of Hyperboloids and Paraboloids | |
9.37.5. | Surfaces of Revolution | |
9.37.6. | Tangent Planes to Parametric Surfaces | |
9.37.7. | Surface Areas of Parametric Surfaces | |
9.37.8. | Surface Areas of Special Surfaces | |
9.37.9. | Surface Areas of Non-Parametric Surfaces |
9.38.1. | Surface Integrals of Scalar Functions Over Parametric Surfaces | |
9.38.2. | Surface Integrals of Scalar Functions Over Non-Parametric Surfaces | |
9.38.3. | Surface Integrals of Vector-Valued Fields | |
9.38.4. | Surface Integrals in Cylindrical Polar Coordinates | |
9.38.5. | Surface Integrals in Spherical Polar Coordinates | |
9.38.6. | The Flux of a Vector Field in Three Dimensions |
9.39.1. | The Divergence Theorem | |
9.39.2. | The Divergence Theorem With Composite Surfaces | |
9.39.3. | Stokes' Theorem | |
9.39.4. | Ampere’s Law | |
9.39.5. | Faraday’s Law of Electromagnetic Induction |