Understanding Multivariable Calculus: Problems, Solutions, and Tips

Course No. 1023
Professor Bruce H. Edwards, Ph.D.
University of Florida
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Course No. 1023
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Course Overview

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

  • 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;
  • new coordinate systems that enable us to solve integrals whose solutions in Cartesian coordinates may be difficult to work with; 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 basic concepts of calculus together in a much deeper and more powerful way. This course is the next step for students and professionals to expand their knowledge for work or study in many quantitative fields, as well as 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 the 36 comprehensive lectures 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.

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.

With these new tools, 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.

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. With graphics animated with state-of-the-art software that brings three-dimensional surfaces and volumes to life, as well as an accompanying illustrated workbook, this course will provide anyone who is intrigued about math a chance to better understand the full potential of one of the crowning mathematical achievements of humankind.

About Your Professor

Dr. Bruce H. Edwards is Professor of Mathematics at the University of Florida. He earned his Ph.D. in Mathematics from Dartmouth College. He has been honored with numerous Teacher of the Year awards as well as awards for his work in mathematics education for the state of Florida.

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36 lectures
 |  Average 30 minutes each
  • 1
    A Visual Introduction to 3-D Calculus
    Review key concepts from basic calculus, then immediately jump into three dimensions with a brief overview of what you’ll be learning. Apply distance and midpoint formulas to three-dimensional objects in your very first of many extrapolations from two-dimensional to multidimensional calculus, and observe some of the new curiosities unique to functions of more than one variable. x
  • 2
    Functions of Several Variables
    What makes a function “multivariable?” Begin with definitions, and then see how these new functions behave as you apply familiar concepts of minimum and maximum values. Use graphics and other tools to observe their interactions with the xy-plane, and discover how simple functions such as y=x are interpreted differently in three-dimensional space. x
  • 3
    Limits, Continuity, and Partial Derivatives
    Apply fundamental definitions of calculus to multivariable functions, starting with their limits. See how these limits become complicated as you approach them, no longer just from the left or right, but from any direction and along any path. Use this to derive the definition of a versatile new tool: the partial derivative. x
  • 4
    Partial Derivatives—One Variable at a Time
    Deep in the realm of partial derivatives, you’ll discover the new dimensions of second partial derivatives: differentiate either twice with respect to x or y, or with respect once each to x and y. Consider Laplace’s equation to see what makes a function “harmonic.” x
  • 5
    Total Differentials and Chain Rules
    Complete your introduction to partial derivatives as you combine the differential and chain rule from elementary calculus and learn how to generalize them to functions of more than one variable. See how the so-called total differential can be used to approximate ?z over small intervals without calculating the exact values. x
  • 6
    Extrema of Functions of Two Variables
    The ability to find extreme values for optimization is one of the most powerful consequences of differentiation. Begin by defining the Extreme Value theorem for multivariable functions and use it to identify relative extrema using a “second partials test”—which you may recognize as a logical extension of the “second derivative test” used in Calculus I. x
  • 7
    Applications to Optimization Problems
    Continue the exploration of multivariable optimization by using the Extreme Value theorem on closed and bounded regions. Find absolute minimum and maximum values across bounded regions of a function, and apply these concepts to a real-world problem: attempting to minimize the cost of a water line’s construction. x
  • 8
    Linear Models and Least Squares Regression
    Apply techniques of optimization to curve-fitting as you explore an essential statistical tool yielded by multivariable calculus. Begin with the Least Squares Regression Line that yields the best fit to a set of points. Then, apply it to a real-life problem by using regression to approximate the annual change of a man’s systolic blood pressure. x
  • 9
    Vectors and the Dot Product in Space
    Begin your study of vectors in three-dimensional space as you extrapolate vector notation and formulas for magnitude from the familiar equations for two dimensions. Then, equip yourself with an essential new means of notation as you learn to derive the parametric equations of a line parallel to a direction vector. x
  • 10
    The Cross Product of Two Vectors in Space
    Take the cross product of two vectors by finding the determinant of a 3x3 matrix, yielding a third vector perpendicular to both. Explore the properties of this new vector using intuitive geometric examples. Then, combine it with the dot product from Lecture 9 to define the triple scalar product, and use it to evaluate the volume of a parallelepiped. x
  • 11
    Lines and Planes in Space
    Turn fully to lines and entire planes in three-dimensional space. Begin by defining a plane using the tools you’ve acquired so far, then learn about projections of one vector onto another. Find the angle between two planes, then use vector projections to find the distance between a point and a plane. x
  • 12
    Curved Surfaces in Space
    Beginning with the equation of a sphere, apply what you’ve learned to curved surfaces by generating cylinders, ellipsoids, and other so-called quadric surfaces. Discover the recognizable parabolas and other 2D shapes that lay hidden in new vector equations, and observe surfaces of revolution in three-dimensional space. x
  • 13
    Vector-Valued Functions in Space
    Consolidate your mastery of space by defining vector-valued functions and their derivatives, along with various formulas relating to arc length. Immediately apply these definitions to position, velocity, and acceleration vectors, and differentiate them using a surprisingly simple method that makes vectors one of the most formidable tools in multivariable calculus. x
  • 14
    Kepler’s Laws—The Calculus of Orbits
    Blast off into orbit to examine Johannes Kepler’s laws of planetary motion. Then apply vector-valued functions to Newton’s second law of motion and his law of gravitation, and see how Newton was able to take laws Kepler had derived from observation and prove them using calculus. x
  • 15
    Directional Derivatives and Gradients
    Continue to build on your knowledge of multivariable differentiation with gradient vectors and use them to determine directional derivatives. Discover a unique property of the gradient vector and its relationships with level curves and surfaces that will make it indispensable in evaluating relationships between surfaces in upcoming lectures. x
  • 16
    Tangent Planes and Normal Vectors to a Surface
    Utilize the gradient to find normal vectors to a surface, and see how these vectors interplay with standard functions to determine the tangent plane to a surface at a given point. Start with tangent planes to level surfaces, and see how your result compares with the error formula from the total differential. x
  • 17
    Lagrange Multipliers—Constrained Optimization
    It’s the ultimate tool yielded by multivariable differentiation: the method of Lagrange multipliers. Use this intuitive theorem and some simple algebra to optimize functions subject not just to boundaries, but to constraints given by multivariable functions. Apply this tool to a real-world cost-optimization example of constructing a box. x
  • 18
    Applications of Lagrange Multipliers
    How useful is the Lagrange multiplier method in elementary problems? Observe the beautiful simplicity of Lagrange multipliers firsthand as you reexamine an optimization problem from Lecture 7 using this new tool. Next, explore one of the many uses of constrained optimization in the world of physics by deriving Snell’s Law of Refraction. x
  • 19
    Iterated integrals and Area in the Plane
    With your toolset of multivariable differentiation finally complete, it’s time to explore the other side of calculus in three dimensions: integration. Start off with iterated integrals, an intuitive and simple approach that merely adds an extra step and a slight twist to one-dimensional integration. x
  • 20
    Double Integrals and Volume
    In taking the next step in learning to integrate multivariable functions, you’ll find that the double integral has many of the same properties as its one-dimensional counterpart. Evaluate these integrals over a region R bounded by variable constraints, and extrapolate the single variable formula for the average value of a function to multiple variables. x
  • 21
    Double Integrals in Polar Coordinates
    Explore integration from a whole new perspective, first by transforming Cartesian functions f(x.y) into polar coordinates defined by r and ?. After getting familiar with surfaces defined by this new coordinate system, see how these coordinates can be used to derive simple and elegant solutions from integrals whose solutions in Cartesian coordinates may be arduous to derive. x
  • 22
    Centers of Mass for Variable Density
    With these new methods of evaluating integrals over a region, we can apply these concepts to the realm of physics. Continuing from the previous lecture, learn the formulas for mass and moments of mass for a planar lamina of variable density, and find the center of mass for these regions. x
  • 23
    Surface Area of a Solid
    Bring another fundamental idea of calculus into three dimensions by expanding arc lengths into surface areas. Begin by reviewing arc length and surfaces of revolution, and then conclude with the formulas for surface area and the differential of surface area over a region. x
  • 24
    Triple Integrals and Applications
    Explore integration from a whole new perspective, first by transforming Cartesian functions f(x.y) into polar coordinates defined by r and ?. After getting familiar with surfaces defined by this new coordinate system, see how these coordinates can be used to derive simple and elegant solutions from integrals whose solutions in Cartesian coordinates may be arduous to derive. x
  • 25
    Triple Integrals in Cylindrical Coordinates
    Just as you applied polar coordinates to double integrals, you can now explore their immediate extension into volumes with cylindrical coordinates—moving from a surface defined by (r,?) to a cylindrical volume with an extra parameter defined by (r,?,z). Use these conversions to simplify problems. x
  • 26
    Triple Integrals in Spherical Coordinates
    Similar to the shift from rectangular coordinates to cylindrical coordinates, you will now see how spherical coordinates often yield more useful information in a more concise format than other coordinate systems—and are essential in evaluating triple integrals over a spherical surface. x
  • 27
    Vector Fields—Velocity, Gravity, Electricity
    In your introduction to vector fields, you will learn how these creations are essential in describing gravitational and electric fields. Learn the definition of a conservative vector field using the now-familiar gradient vector, and calculate the potential of a conservative vector field on a plane. x
  • 28
    Curl, Divergence, Line Integrals
    Use the gradient vector to find the curl and divergence of a field—curious properties that describe the rotation and movement of a particle in these fields. Then explore a new, exotic type of integral, the line integral, used to evaluate a density function over a curved path. x
  • 29
    More Line Integrals and Work by a Force Field
    One of the most important applications of the line integral is its ability to calculate work done on an object as it moves along a path in a force field. Learn how vector fields make the orientation of a path significant. x
  • 30
    Fundamental Theorem of Line Integrals
    Generalize the fundamental theorem of calculus as you explore the key properties of curves in space as they weave through vector fields in three dimensions. Then find out what makes a curve smooth, piecewise-smooth, simple, and closed. Next, manipulate curves to reveal new, simpler methods of evaluating some line integrals. x
  • 31
    Green’s Theorem—Boundaries and Regions
    Using one of the most important theorems in multivariable calculus, observe how a line integral can be equivalent to an often more-workable area integral. From this, you will then see why the line integral around a closed curve is equal to zero in a conservative vector field. x
  • 32
    Applications of Green’s Theorem
    With the full power of Green’s theorem at your disposal, transform difficult line integrals quickly and efficiently into more approachable double integrals. Then, learn an alternative form of Green’s theorem that generalizes to some important upcoming theorems. x
  • 33
    Parametric Surfaces in Space
    Extend your understanding of surfaces by defining them in terms of parametric equations. Learn to graph parametric surfaces and to calculate surface area. x
  • 34
    Surface Integrals and Flux Integrals
    Discover a key new integral, the surface integral, and a special case known as the flux integral. Evaluate the surface integral as a double integral and continue your study of fluid mechanics by utilizing this integral to evaluate flux in a vector field. x
  • 35
    Divergence Theorem—Boundaries and Solids
    Another hallmark of multivariable calculus, the Divergence theorem, combines flux and triple integrals, just as Green’s theorem combines line and double integrals. Discover the divergence of a fluid, and call upon the gradient vector to define how a surface integral over a boundary can give the volume of a solid. x
  • 36
    Stokes’s Theorem and Maxwell's Equations
    Complete your journey by developing Stokes’s theorem, the third capstone relationship between the new integrals of multivariable calculus, seeing how a line integral equates to a surface integral. Conclude with connections to Maxwell’s famous equations for electric and magnetic fields—a set of equations that gave birth to the entire field of classical electrodynamics. x

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Your professor

Bruce H. Edwards

About Your Professor

Bruce H. Edwards, Ph.D.
University of Florida
Dr. Bruce H. Edwards is Professor of Mathematics at the University of Florida. Professor Edwards received his B.S. in Mathematics from Stanford University and his Ph.D. in Mathematics from Dartmouth College. After his years at Stanford, he taught mathematics at a university near Bogot·, Colombia, as a Peace Corps volunteer. Professor Edwards has won many teaching awards at the University of Florida, including Teacher of...
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Understanding Multivariable Calculus: Problems, Solutions, and Tips is rated 4.9 out of 5 by 34.
Rated 5 out of 5 by from Excellent Communicator Purchased this title a month ago. I’m one third of the way through it and am exceptionally pleased. The instructor communicates well with clear explanations. Lectures are accompanied by professional 3D animated graphics.
Date published: 2019-06-28
Rated 4 out of 5 by from Well thought out expostion I am now over half way through the course. Professor Edwards is a good teacher, although a bit stronger on showing the calculations than making some of the logic fully clear. However, this seems to be mainly the result of his being required to read out every mathematical expression - boring for him, very tedious for the watcher. Given the viewer is likely to have watched Calculus I and II, at least Professor Edwards assumes that, some lines of math could just be shown and not painfully read out. I fear this is because the course is also provided in audio, but I cannot imagine that any audio course of this subject can be of any benefit to the student, better to recognize that it has to be video and spent more time on motivation and explanation and less on reading out every term of the mathematics.
Date published: 2019-04-19
Rated 5 out of 5 by from This was a wonderful experience in all ways for anyone with an interest in higher math. I also took the first 2 calc. courses by this professor. I cannot praise his presentations enough! This course will add another dimension to your math knowledge-the 3rd dimension-maybe the most important! My only question revolves around the math solution for the hollowed out sphere section as I get 3 to the three halves power as being about 5.196-can anyone help send me an e-mail if you can clarify the math.
Date published: 2018-09-02
Rated 4 out of 5 by from Good, but not great. This is a wonderful course on multi-variable calculus, generalizing elementary calculus to higher dimensions. It has an applied slant to it, focusing more on cookbook computations rather than underlying theory and concepts. At an introductory level it is ideal for engineers or scientists that apply these ideas to their field. Let me start off with the negatives, which is why the course gets only 4 stars instead of 5. First of all I’ve completed Edwards’ three part Calculus sequence, and this is by far the weakest title in the trilogy. The reason for this is many important results are glossed over, and are not built up like they are in the previous titles. The course starts off great, in the spirit of its predecessors, as with an interesting discussion in Lecture 8 as Edwards discusses some of his own research experience about an interesting result he came up with by playing with values. The highlight of this early sequence is lecture 12, where quadratic surfaces are each discussed in detail with ample visualizations. Then things begin to turn sour. Lecture 14 is the start of the downfall, a completely useless lecture that tries to cover Kepler’s Laws but simply hand waives everything and gives problems that are completely unrelated to everything in the sequence so far(the previous titles did a much better job on this topic). There is a good and simple development of the gradient and directional derivatives in lecture 15 with a proof why of why it points in the direction of maximum increase, but by the next lecture 16 things continue downhill. Lecture 16 simply fails to explain and elaborate on a very important concept, that the gradient is normal to its level curve/surface (and in fact the function remains constant in a direction normal to the gradient – this is not even mentioned!). This important concept is superficially stated when a simple two minute argument would have helped internalize it instead of working more examples. Some pictures of gradient fields that show how they behave near extrema would have been very helpful. Next, two lectures (17&18) are devoted to Lagrange multipliers. But in two whole lectures the idea is never truly developed, only more examples are worked. A simple geometric argument related to the normal and gradient illustrates why the method works but this is never discussed. Multi-variable integration is then developed in a rather mediocre way. There is an overemphasis on viewing double integrals as area before seeing them in their more natural form as volume, and the whole notion of “iterated” is both poorly explained and illustrated. Regions of integration just appear out of the blue and I don’t believe are very well explained nor are visuals taken advantage of to drive home the point. In lecture 22 center of mass is sloppily developed (compare this to how well center of mass was treated in the previous title, where everything was developed from the ground up). Lecture 23 links the notion of surface area to arc length, but the formula is never derived – the professor simply brushes it off as “another calculus derivation” depriving the viewer the opportunity to see how multi-variable derivations are made and applying previous concepts in a novel way (a three minute derivation would have done more than another mindless example). At least in lecture 26 spherical coordinates are derived, but when it comes to deriving the volume element they again are dismissed in favor of another mindless example, where a few minutes would have sufficed and provided more understanding. This trend of “here is the formula, now lets solve 4 examples” continues until lecture 28. Curl and divergence are simply given as formulas, where Edwards at least mentions what they actually mean with some field examples, but the formulas are never explained or derived and mysteriously appear out of the blue. I’m not asking for fully rigorous proofs, but a very simple explanation of the intuition behind the results would not have taken much time and would really help internalize the concept. From lecture 29 on, things get a lot better. After walking you through line integrals, you get a solid development that leads to Green’s Theorem and the best lecture in the course culminates in lecture 32 where it is applied. Here Edwards is at his greatest again, giving a fascinating result on how to compute the area of a polygon. The course concludes with surface integrals, and the divergence and Stoke’s theorems, and while these results won’t immediately be appreciated or deeply understood (because all previous notions are not thoroughly developed, but also because the details belong in a vector analysis course) they at least provide computation practice and are connected to the well developed Green's Theorem. What is great is how all the three major theorems are compared and contrasted to the FTC which relates an inner region to some boundary. In conclusion, I will note that there are not that many great resources for multi-variable calculus in general, so this one is still much better and a student has a lot to gain from completing the course. While I understand this is supposed to be an introductory survey of the field, I feel Edwards was not at his best here compared to his earlier titles. Maybe he was burned out after recording those two other courses, or he simply does not have as much experience teaching multi-variable calculus as he does elementary calculus. But I felt this title was exceedingly superficial at some crucial points, where a simple geometrical argument instead of another mindless example would have clarified concepts. Presumably students taking this course had already completed elementary calculus so they didn’t need to see simple integrals re-evaluated instead of new ideas being elaborated upon. He was not shy in exposing students to trigonometric substitution and complicated series in the prequels, so I don’t see why Edwards felt like talking down to students or being lazy and telling them to “look in a textbook” - he should not have been afraid to challenge students who are already at this level. For this reason you might need Khan Academy or other Youtube videos to fill in the gaps, especially with div and curl. On a separate note, the guide book was indispensable, and provided good drill exercises (with solutions). In summary: a worthwhile course overall, but needlessly superficial at some crucial points when it needed not be.
Date published: 2018-08-01
Rated 5 out of 5 by from Great teacher - The material was vaguely familiar from Caltech, but easier this time around to understand - I was a biology major! The pictures are good, and helped me to "see" what was happening. I got it on a bet, that I couldn't do it this time either! Not so!!
Date published: 2018-07-28
Rated 5 out of 5 by from Great calculus course I took multivariable calculus many years ago in college, but then forgot everything. I found this course to be exceptionally clear. The examples are engaging. The workbook reinforces the necessary skills without being overly burdensome. If you need a calculus refresher or know someone taking multivariable calculus now, get this course! It would be helpful to engineering or physics students too.
Date published: 2018-06-29
Rated 5 out of 5 by from Stellar Course I highly recommend this course to those who enjoy mathematics and want to get a good understanding of multivariable calculus. It is the best math course I’ve ever taken. Professor Edwards has produced excellent lectures that are clear and understandable, along with problems in the work book that are challenging and which reinforce the lectures. The visual aids are superb. Students should preferably be proficient with basic calculus before taking this course, although Professor Edwards does review fundamental concepts before employing them. While a graphing calculator is not necessary for understanding the material, having access to one, particularly one with 3D capabilities, would be helpful. My studying a chapter, seeing the lecture, and working all of the work book problems for that chapter typically took 2 to 3 hours. I highly recommend doing the problems as a major part of the learning process; Professor Edwards provides answers to the problems. I hope The Great Courses asks Professor Edwards to do additional mathematics courses for them such as Linear Algebra.
Date published: 2017-11-28
Rated 5 out of 5 by from Professional Solution. More fun than a barrel of adjectives! A fresh attitude for an old friend, with a blast of encouragement to keep the day interesting.
Date published: 2017-10-24
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