Understanding Multivariable Calculus: Problems, Solutions, and Tips

Course Overview

Gain a profound understanding of multivariable calculus with this excellent and clear guide that is useful for students, professionals, and lovers of mathematics. This 36-lecture course, taught by award-winning Professor Bruce Edwards of the University of Florida, brings the basic concepts of calculus together in a deeper and more powerful way. Designed for anyone familiar with basic calculus, this course focuses on deepening and generalizing fundamental tools of integration and differentiation to functions of more than one variable.

Professor Edwards leads you through new techniques with clarity and enthusiasm, making even the most challenging material accessible and enjoyable through animated graphics and an illustrated workbook.

Video Lectures

1. A Visual Introduction to 3-D Calculus – 34 min
   Review key concepts from basic calculus and apply distance and midpoint formulas to three-dimensional objects.
2. Functions of Several Variables – 30 min
   Learn definitions and behaviors of multivariable functions and their interactions with the xy-plane.
3. Limits, Continuity, and Partial Derivatives – 30 min
   Apply fundamental definitions to multivariable functions starting with limits and derive partial derivatives.
4. Partial Derivatives-One Variable at a Time – 30 min
   Discover second partial derivatives and what makes a function “harmonic” with Laplace’s equation.
5. Total Differentials and Chain Rules – 31 min
   Generalize the differential and chain rule to multivariable functions.
6. Extrema of Functions of Two Variables – 31 min
   Define the Extreme Value theorem and find relative extrema using a “second partials test”.
7. Applications to Optimization Problems – 31 min
   Use the Extreme Value theorem on closed and bounded regions to find optimal solutions to real-world problems.
8. Linear Models and Least Squares Regression – 31 min
   Apply optimization techniques to curve-fitting and understand the Least Squares Regression Line.
9. Vectors and the Dot Product in Space – 30 min
   Begin studying vectors in three-dimensional space and derive parametric equations for lines.
10. The Cross Product of Two Vectors in Space – 29 min
    Explore the properties of the cross product and define the triple scalar product.
11. Lines and Planes in Space – 32 min
    Define planes using vector tools and find distances between points and planes.
12. Curved Surfaces in Space – 31 min
    Generate cylinders and ellipsoids, and discover surfaces of revolution.
13. Vector-Valued Functions in Space – 31 min
    Define vector-valued functions and their derivatives, and apply them to position, velocity, and acceleration.
14. Kepler’s Laws-The Calculus of Orbits – 30 min
    Examine planetary motion and apply calculus to Newton’s laws.
15. Directional Derivatives and Gradients – 30 min
    Determine directional derivatives using gradient vectors and explore their relationships.
16. Tangent Planes and Normal Vectors to a Surface – 29 min
    Find normal vectors to surfaces and determine tangent planes at points.
17. Lagrange Multipliers-Constrained Optimization – 31 min
    Use Lagrange multipliers for optimizing functions under constraints.
18. Applications of Lagrange Multipliers – 30 min
    Explore the use of Lagrange multipliers in physics for deriving Snell’s Law.
19. Iterated Integrals and Area in the Plane – 30 min
    Start exploring integration with iterated integrals in three dimensions.
20. Double Integrals and Volume – 30 min
    Learn to evaluate double integrals over regions bounded by variable constraints.
21. Double Integrals in Polar Coordinates – 31 min
    Transform Cartesian functions to polar coordinates for simpler integral solutions.
22. Centers of Mass for Variable Density – 30 min
    Apply concepts of mass and moments for regions of variable density.
23. Surface Area of a Solid – 31 min
    Expand arc lengths into surface areas and review formulas for surface area.
24. Triple Integrals and Applications – 29 min
    Use triple integrals to find the volume of solids in space.
25. Triple Integrals in Cylindrical Coordinates – 31 min
    Explore triple integrals in cylindrical coordinates for easier problem-solving.
26. Triple Integrals in Spherical Coordinates – 30 min
    Use spherical coordinates to evaluate triple integrals over spherical surfaces.
27. Vector Fields-Velocity, Gravity, Electricity – 30 min
    Understand vector fields in describing gravitational and electric fields.
28. Curl, Divergence, Line Integrals – 31 min
    Define curl and divergence, and explore line integrals used for evaluating density functions.
29. More Line Integrals and Work by a Force Field – 31 min
    Calculate work done on objects moving along paths in force fields.
30. Fundamental Theorem of Line Integrals – 31 min
    Generalize the fundamental theorem of calculus and evaluate line integrals in vector fields.
31. Green’s Theorem-Boundaries and Regions – 31 min
    Learn how line integrals relate to area integrals and properties of conservative vector fields.
32. Applications of Green’s Theorem – 30 min
    Transform line integrals into double integrals using Green’s theorem.
33. Parametric Surfaces in Space – 32 min
    Graph parametric surfaces and calculate surface area.
34. Surface Integrals and Flux Integrals – 31 min
    Discover surface integrals and evaluate flux in vector fields.
35. Divergence Theorem-Boundaries and Solids – 29 min
    Combine concepts of flux and triple integrals to define volumes.
36. Stokes’s Theorem and Maxwell’s Equations – 34 min
    Conclude with Stokes’s theorem and its relation to Maxwell’s equations in classical electrodynamics.