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# Multivariable Calculus

Generalize your understanding of calculus to vector-valued functions and functions of multiple variables.

## Content

### 2D and 3D Space

• Construct equations of lines and planes.
• Classify and analyze quadric surfaces.
• Describe regions and transformations of regions in space using formal terminology.
• Convert between rectangular, cylindrical, and spherical coordinates.
• Visualize and geometrically describe the level surfaces of a function.

### Partial Derivatives

• Extend prior knowledge of single-variable derivative rules to compute partial derivatives of multivariable functions, including the chain rule and implicit differentiation.
• Use gradient notation to express the multivariable chain rule more concisely and compute directional derivatives.
• Compute differential geometry features associated with space curves and surfaces including tangents, curvature, arc length, and osculating plane.
• Use differentials and linearization to approximate values of multivariable functions.

### Optimization

• Extend prior knowledge of single-variable optimization to perform constrained optimization with multivariable functions in real-world modeling scenarios.
• Use Lagrange multipliers as an alternative method to optimize a multivariable function.

### Double and Triple Integrals

• Evaluate Riemann sums of multivariable functions and interpret them geometrically.
• Compute the integral of a multivariable function as the limit of a Riemann sum.
• Extend prior knowledge of single-variable integrals to evaluate double and triple integrals using the fundamental theorem of calculus.
• Solve advanced integrals by performing change of variables into a non-rectangular coordinate system.

### Vector Fields

• Sketch the gradient vector field for a given function.
• Determine whether a given vector field is conservative, and in the event that it is, compute a potential function for it.
• Compute and interpret divergence, curl, circulation, and flux of vector fields.

### Line and Surface Integrals

• Compute line integrals over a variety of curves.
• Use the fundamental theorem of line integrals to compute a line integral when the associated vector field is conservative.
• Understand how the fundamental theorem of line integrals implies path independence of conservative vector fields.
• Use Green’s theorem to evaluate line integrals and compute areas.
• Find parametrizations and compute surface areas of parametric surfaces.
• Evaluate surface integrals in a variety of coordinate systems.
• Use the divergence theorem and Stokes’ theorem to convert between single, double, and triple integrals when doing so eases computations.

### Applications to Physics

• Use partial derivatives to solve problems in vector mechanics.
• Use integration to calculate physical quantities for plane regions and wires and work done by a force along a curve.
1.
Vector Functions and Vector Fields
27 topics
1.1. Vector-Valued Functions
 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. Curvature
 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. Vector Fields
 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. Divergence and Curl
 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.
Multivariable Functions
50 topics
2.5. Introduction to Multivariable Functions
 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. Partial Derivatives
 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. The Multivariable Chain Rule
 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. Plane Transformations
 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. Differentiation
 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. Optimization
 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.
Multiple Integrals
36 topics
3.13. Riemann Sums
 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. Double Integrals
 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. Triple 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. Change of Variables for Double Integrals
 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. Change of Variables for Triple Integrals
 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.
Line Integrals
30 topics
4.18. Line Integrals of Scalar Functions
 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. Line Integrals of Scalar Functions Over Parametric Curves
 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. Line Integrals With Respect to X and Y
 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. Line Integrals of Vector-Valued Functions
 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. Circulation and Flux
 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. Green's Theorem
 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.
Surface Integrals
21 topics
5.24. Parametric Surfaces
 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. Surface Area
 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. Surface Integrals
 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.
Applications of Multivariable Calculus
21 topics
6.27. Vector Mechanics
 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. Applications of Multiple Integrals
 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. Random Variables
 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