Understanding Multivariable Calculus: Problems, Solutions, and Tips [repost]

Calculus offers some of the most astounding advances in all of mathematics—reaching far beyond the two-dimensional applications learned in first-year calculus. We do not live on a sheet of paper, and in order to understand and solve rich, real-world problems of more than one variable, we need multivariable calculus, where the full depth and power of calculus is revealed.

Whether calculating the volume of odd-shaped objects, predicting the outcome of a large number of trials in statistics, or even predicting the weather, we depend in myriad ways on calculus in three dimensions. Once we grasp the fundamentals of multivariable calculus, we see how these concepts unfold into new laws, entire new fields of physics, and new ways of approaching once-impossible problems.

With multivariable calculus, we get

partial derivatives, which are the building blocks of partial differential equations;
new tools for optimization, taking into account as many variables as needed;
vector fields that give us a peek into the workings of fluids, from hydraulic pistons to ocean currents and the weather;
line integrals that determine the work done on a path through these vector fields;
new coordinate systems that enable us to solve integrals whose solutions in Cartesian coordinates may be difficult to work with;
force fields, including those of gravitation and electricity; and
mathematical definitions of planes and surfaces in space, from which entire fields of mathematics such as topology and differential geometry arise.

Understanding Multivariable Calculus: Problems, Solutions, and Tips, taught by award-winning Professor Bruce H. Edwards of the University of Florida, brings the concepts of calculus together in a much deeper and more powerful way. Building from an understanding of basic concepts in Calculus I, it is a full-scope course that encompasses all the key topics of multivariable calculus, together with brief reviews of needed concepts as you go along. This course is the next step for students and professionals to expand their knowledge for work or study in mathematics, statistics, science, or engineering and to learn new methods to apply to their field of choice. It’s also an eye-opening intellectual exercise for teachers, retired professionals, and anyone else who wants to understand the amazing applications of 3-D calculus.

Designed for anyone familiar with basic calculus, Understanding Multivariable Calculus follows, but does not essentially require knowledge of, Calculus II. The few topics introduced in Calculus II that do carry over, such as vector calculus, are here briefly reintroduced, but with a new emphasis on three dimensions.

Your main focus throughout, in a series of 36 comprehensive lectures that extend beyond what is typically taught in university classrooms, is on deepening and generalizing fundamental tools of integration and differentiation to functions of more than one variable. Under the expert guidance of Professor Edwards, you’ll embark on an exhilarating journey through the concepts of multivariable calculus, enlivened with real-world examples and beautiful animated graphics that lift calculus out of the textbook and into our three-dimensional world.

Unlock the Full Power of Calculus

Too frequently, students end their education in higher-level math after a year of fundamental calculus–thereby missing out on this capstone course that makes possible consideration of problems that have the dimensions and complexity of real life. With the tools and techniques of multivariable calculus, students will be able to understand and solve complex problems arising in a wide array of fields in an elegant manner.

Use the gradient vector to optimize a function subject to a constraint using Lagrange multipliers.
Determine work done on a path with line integrals, and find the flow through a surface with surface integrals.
See how Isaac Newton used multivariable calculus to prove Johannes Kepler’s laws of orbits.
Explore the properties of fluids and the relationship of vector fields with path integrals using Stokes’s theorem, Green’s theorem, and the Divergence theorem.
Understand why Maxwell was able to discover underlying unities between electricity and magnetism that no one had been able to identify before.

In Understanding Multivariable Calculus, you will begin your journey in familiar territory as you jump into three dimensions with standard Cartesian coordinates. You’ll see the fundamentals you learned in Calculus I extrapolated to three dimensions, as derivatives and the Extreme Value theorem are applied with more variables. Then observe how an extra dimension enables partial derivatives to provide new information, including how they can be combined into a total differential that enables changes in a multivariable function to be approximated with its partial derivatives.

By adding this extra dimension to vectors, you will be given surprising new insight into surfaces and volumes from perspectives you never considered. You’ll view how vectors can be combined with recognizable techniques from geometry and algebra to yield parametric equations that are both powerful and simple in defining lines and planes in space.

Next, you’ll see old integrals in a new light, as two definite integrals with two different variables are computed as an iterated integral. By tweaking this method, you’ll learn to compute double integrals for area, as well as triple integrals for volume—as well as some helpful techniques involving basic algebra and new coordinate systems for setting up your integrals for success.

Finally, witness a truly wonderful thing happen when these new double and triple integrals are combined with what you’ve learned in previous lectures about vectors and derivatives: Entire new fields of physics explode into existence. Watch as line integrals, surface integrals, curl, divergence, and flux are derived and illuminate fluid mechanics. Then you’ll see how the famous theorems of Green and Stokes and the Divergence theorem unite these integrals, and be granted insight into how Maxwell derived his equations that gave birth to the unified study of electromagnetism.

A New Look at Old Problems

How do you integrate over a region of the xy plane that can’t be defined by just one standard y = f(x) function? Multivariable calculus is full of hidden surprises, containing the answers to many such questions. In Understanding Multivariable Calculus, Professor Edwards unveils powerful new tools in every lecture to solve old problems in a few steps, turn impossible integrals into simple ones, and yield exact answers where even calculators can only approximate.

And with techniques using new coordinate systems, new integrals, and new theorems uniting them all, you will be able to integrate volumes and surface areas directly with double and triple integrals;
define easily differentiable parametric equations for a function using vectors; and
utilize polar, cylindrical, and spherical coordinates to evaluate double and triple integrals whose solutions are difficult in standard Cartesian coordinates.

These tools are essential in fields such as statistics, engineering, and physics where equations arise that cannot be worked with easily using the conventional Cartesian coordinate system.

Professor Edwards leads you through these new techniques with a clarity and enthusiasm for the subject that make even the most challenging material accessible and enjoyable. From the very first lecture, you’ll see why Professor Edwards has won teaching awards at the University of Florida.

Discover at Your Own Pace

With 36 lectures featuring graphics animated with state-of-the-art software that brings three-dimensional surfaces and volumes to life, this course will provide you with a view of multivariable calculus beyond what’s available in textbooks and lecture halls. Using the accompanying illustrated workbook, you are free to move at your own pace to grasp the powerful tools of multivariable calculus to your own satisfaction.

This course offers a uniquely self-contained approach, appealing to a wide array of backgrounds and experience levels. Understanding Multivariable Calculus offers students and professionals in virtually every quantitative field as well as anyone who is intrigued about math a chance to better understand the full potential of one of the crowning mathematical achievements of humankind: calculus.



Lectures:

01 A Visual Introduction to 3-D Calculus
02 Functions of Several Variables
03 Limits, Continuity, and Partial Derivatives
04 Partial Derivatives—One Variable at a Time
05 Total Differentials and Chain Rules
06 Extrema of Functions of Two Variables
07 Applications to Optimization Problems
08 Linear Models and Least Squares Regression
09 Vectors and the Dot Product in Space
10 The Cross Product of Two Vectors in Space
11 Lines and Planes in Space
12 Curved Surfaces in Space
13 Vector-Valued Functions in Space
14 Kepler’s Laws—The Calculus of Orbits
15 Directional Derivatives and Gradients
16 Tangent Planes and Normal Vectors to a Surface
17 Lagrange Multipliers—Constrained Optimization
18 Applications of Lagrange Multipliers
19 Iterated integrals and Area in the Plane
20 Double Integrals and Volume
21 Double Integrals in Polar Coordinates
22 Centers of Mass for Variable Density
23 Surface Area of a Solid
24 Triple Integrals and Applications
25 Triple Integrals in Cylindrical Coordinates
26 Triple Integrals in Spherical Coordinates
27 Vector Fields—Velocity, Gravity, Electricity
28 Curl, Divergence, Line Integrals
29 More Line Integrals and Work by a Force Field
30 Fundamental Theorem of Line Integrals
31 Green’s Theorem—Boundaries and Regions
32 Applications of Green’s Theorem
33 Parametric Surfaces in Space
34 Surface Integrals and Flux Integrals
35 Divergence Theorem—Boundaries and Solids
36 Stokes’s Theorem and Maxwell's Equations