# Multivariable Calculus

Our multivariable course provides in-depth coverage of the calculus of vector-valued and multivariable functions, vector fields, multiple integrals, line and surface integrals, and their real-world applications. This comprehensive course will prepare students for further studies in advanced mathematics, engineering, statistics, machine learning, and other fields requiring a solid foundation in multivariable calculus.

## Content

Students enhance their understanding of vector-valued functions to include analyzing limits and continuity with vector-valued functions, applying rules of differentiation and integration, unit tangent, principal normal and binormal vectors, osculating planes, parametrization by arc length, and curvature.

As part of this course, students undertake a deep dive into the differential calculus of multivariable functions, including partial derivatives, tangent planes, linear approximations, the gradient vector, directional derivatives, the multivariable chain rule, and differentials. Students then generalize their understanding of derivatives of multivariable and vector-valued functions to include the total derivative of a function, second derivatives, and multivariable Taylor polynomials.

Students develop their knowledge of optimization techniques to include unconstrained optimization of multivariable functions and the role of Taylor polynomials and the Hessian determinant in classifying critical points. Students will also solve constrained optimization problems using the method of Lagrange multipliers.

This course extends students' understanding of integration to multiple integrals, including their formal construction using Riemann sums, calculating multiple integrals over various domains, and applications of multiple integrals. Students explore affine and nonlinear transformations in the plane and use this to understand the role of the Jacobian in performing changes of variables with multiple integrals, including cylindrical and spherical polar coordinates.

The course concludes with an exploration of vector fields, line integrals for scalar and vector-valued functions, the fundamental theorem of line integrals, circulation and flux in two-dimensional and three-dimensional vector fields, Green's theorem, surface integrals, Stokes' theorem, and the divergence theorem.

Upon successful completion of this course, students will have mastered the following:
• Extend previous understandings of vector-valued functions to include functions with three spatial components and compute their domains, limits, derivatives, and integrals, and describe conditions for continuity and differentiability of these functions.
• Analyze space curves, including understanding and computing unit tangent, principal normal, and binormal vectors, osculating planes, and parameterizing a curve by arc length.
• Understand curvature and radius of curvature, and compute these quantities for various curves.
• Learn essential concepts related to multivariable functions, including their various representations, domains, ranges, limits, and continuity.
• Investigate important quadric surfaces and learn how to compute properties such as intersections, traces, and projections of these surfaces onto various planes.
• Compute partial derivatives, interpret their meaning geometrically, and apply Clairaut's theorem.
• Find tangent planes to multivariable functions, and understand how these relate to linear approximations.
• Understand the role of the gradient vector, calculate gradient vectors, and interpret them appropriately. Understand how the gradient vector relates to the directional derivative.
• Use the gradient vector to compute tangent and normal lines and planes to level curves and surfaces.
• Apply the multivariable chain rule, including its extension to polar coordinates, implicit differentiation, and differentials.
• Compute derivatives of multivariable and vector-valued functions, understand their role in forming linear approximations, and their relationship with the gradient vector.
• Compute second derivatives of multivariable functions.
• Form quadratic Taylor polynomials of multivariable functions in scalar and vector form, and use these to form quadratic approximations of functions.
• Understand the definitions of global and local extrema of multivariable functions.
• Classify the critical points of unconstrained multivariable functions using their Taylor polynomials and the second partial derivatives test. Understand how the second partial derivatives test relates to quadratic forms.
• Master the method of Lagrange multipliers for one and multiple constraints.
• Describe the effects of affine and nonlinear transformations on objects in the plane.
• Calculate the Jacobian determinant of a transformation and use it to find critical points.
• Approximate volumes using Riemann sums and construct double integrals using lower and upper bounds.
• Calculate double integrals over rectangular domains, Type I, and Type II regions, and change the order of integration.
• Apply the change of variables formula for double integrals, including converting to plane polar coordinates.
• Calculate triple integrals over rectangular domains, Type I, Type II, and Type III regions, and change the order of integration.
• Master change of variables for triple integrals, including converting to cylindrical polar and spherical polar coordinates.
• Calculate the divergence and curl of a vector field, and describe qualitatively and quantitatively the effects these quantities describe.
• Understand gradient, conservative, and source-free vector fields, and calculate potential and stream functions.
• Compute line integrals of scalar functions in various forms, and understand their properties.
• Compute line integrals of vector-valued functions and understand how these can be used to find circulation and flux in two-dimensional vector fields.
• Apply the fundamental theorem of line integrals, and understand path independence.
• Master Green's theorem conceptually and computationally, including Green's theorem in flux form and methods of extension.
• Understand parametric surfaces, and parametrize quadric and other surfaces.
• Compute surface areas of surfaces and surface integrals of scalar-valued functions.
• Compute the flux of a vector field through a surface.
• Apply Stokes' theorem and relate it to Green's theorem in circulation form.
• Understand and apply the divergence theorem, and relate it to Green's theorem in flux form.
• Apply multivariable calculus concepts to real-world problems in physics, engineering, statistics, and other fields. These include calculating quantities such as mass and charge using density functions, centers of mass, moments of inertia, radii of gyration, finding the average value of a function, and computing probabilities using joint probability density functions.
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