Understanding Multivariable Calculus: Problems, Solutions, and Tips

Rated 5 out of 5 by from Great professor very few them around I'm learning cal 3 before the other two calculus he made it very clear and interesting he is one of hand full prof who could really teach the subject
Date published: 2020-07-23
Rated 5 out of 5 by from A Difficult Subject Explained by a Pro I loved advanced math when I attended college, but confess that my last calculus course was some 42 years ago. I bought both Understanding Calculus II and Understanding Multivariable Calculus and loved them both. Professor Edwards has a way of making calculus not only easy to understand, but interesting. I especially loved it when he said, after completing a problem, "let's savor this for a moment." You knew at that point in the lecture that he was about to really illuminate what was meant by the problem's solution. Great job! Ideal for those beginning on their journey with calculus or those trying to see how much they remembered from so many years ago.
Date published: 2020-07-15
Rated 5 out of 5 by from Excellent Instructor The most remarkable about this excellent course is an excellent teacher. Unlike most advanced math teachers he teaches the student instead of trying to impress the student with how much he, himself, knows..
Date published: 2020-06-18
Rated 5 out of 5 by from First in Class I first studied this subject as an engineering student at Cornell University more than half a century ago. The instruction was passable, the textbook was mediocre and I finished the course with only a scanty knowledge of line and surface integrals, divergence and curl. As for Green’s theorem, I never really ‘got it.’ If I only had Prof. Edwards course back then! This thirty-six-lecture series is a marvel of organization, presentation and clarity. The accompanying course workbook is a gem. It will be a long, long time before another mathematics instructor dares to offer another version of this essential subject. I have sat through about a dozen of ‘The Great Courses.’ None has been better that this one.
Date published: 2020-05-12
Rated 5 out of 5 by from Very Clear Exposition Bruce Edwards is a truly amazing lecturer. I have never eperienced anyone who can present a subject like calculus as clearly as he can. This course is a valuable supplement to any textbook on the subject.
Date published: 2020-02-04
Rated 5 out of 5 by from Good Overview of Calculus Using 3-Dimensions The lecture on the least-squares regression line showed an elegant derivation using partial derivatives. Its simplicity compared to using an algebraic derivation that I was taught to use was very eye opening. The professor keeps all his discussions clear, simple and to the point. I found the course easy to follow.
Date published: 2019-12-27
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
Rated 5 out of 5 by from Multivariable Calculus Sorted. A wonderfully supreme course masterly taught and presented. The animations, graphics, video clips and visuals were magnificent. One of your best courses ever. Very highly recommended.
Date published: 2017-10-09
Rated 4 out of 5 by from Great lecture sequence and examples I purchased mainly as a review/refresher course. The professors presentation is well done with practical (real world) examples. My only minor gripe would be that more time should be devoted to "Physics" concepts like Maxwell's Equations.
Date published: 2017-06-19
Rated 5 out of 5 by from Bruce Edward's teaching. I have purchased Edward's pre-Calculus course as well as one of his calculus courses. His coverage is excellent and his presentations are excellent. In addition, his pedagogy (overall) is superb. I have one complaint. To wit, he refuses to call the natural logarithm by its proper name of "ln" - pronounced "lin". He correctly states that mathematicians and scientists almost always use logs to the base 10 or e, when logarithms can be taken to ANY real number base, except zero or one. He correctly refers to base 10 logarithms as "log", and logarithms to other bases as (for instance) "log to the base 4" of some real number. The base 10 is special and omnipresent so it is proper to refer to them as simply "logs". Well, the base e is also special, and he should be verbally calling logarithms to the base e as "lin" - referring to their mathematics abbreviation, ln. Edwards is the best math teacher I have ever run across. Please tell him I said this, and further add that if he were just a LITTLE bit better, he might be able to teach at Florida State. (He'll like that!) James H. Bentley, PhD
Date published: 2017-02-10
Rated 5 out of 5 by from Good Overview I needed to refresh some topics from my sophomore calculus course, especially div, curl, Green's Theorem, etc., and Prof. Edwards course was perfect for that purpose. The presentation is top-notch, topics are nicely explained, examples and exercises are straightforward. The emphasis is on understanding basic concepts and facility with calculations, not theory per se. In that respect, it would not be a substitute for a 2nd year calculus course, which would spend additional time on proofs, theory, and properties of operators, etc. But theory and proofs can come later - if you want. The exercises in the accompanying study book are generally easy - just what you need to verify your understanding, without getting mired down with the intricacies of very hard problems. Overall, this is a great introductory/refresher course in applying the concepts of multivariate calculus and is well worth the price. I would love to have had this DVD during the summer before my sophomore year in college - with the basic calculations/techniques in place, I would then have been able to better focus on the nuances of theory. Unfortunately, DVDs and the Internet didn't exist back then, but at least today's students have access to this wonderful resource.
Date published: 2016-11-27
Rated 5 out of 5 by from Tough Topic - Terrificallly Taught Going back to school in engineering. This is an advanced topic with many complexities but Prof Edwards does and excellent job explaining the material very clearly and in an organized fashion. Makes it possible for me to keep up with the subject even though has been a long time since I had last covered the topic. Highly recommend this presentation and teacher.
Date published: 2016-06-01
Rated 5 out of 5 by from I liked very much the application on Maxwell’s eq. After completing the 3 courses of calculus given by Professor Edwards, I would like to say that these are the best courses on calculus that I had ever taken. This last course has fantastic visual aids that facilitate the understanding of calculus in 3D. I would strongly suggest to the Teaching Company to get a Lineal Algebra course (by Professor Edwards) to complement these math courses.
Date published: 2016-05-18
Rated 5 out of 5 by from Third is challenging and a charm This is my third TTC calculus course taught by Dr. Edwards. The third is challenging and a charm! Why a charm? This course solidifies fundamental precalculus and elementary calculus skills. Dr. Edwards, once again, emphasizes the necessity for expertise in these fundamentals to understand and enjoy higher mathematics. For me, use of these skills became second nature in understanding and completing these lectures. Calculus III is challenging. This is not a "sit and listen" course; rather it is a course that requires time and engagement. For each lecture, I spent one hour or more to complete the lecture with notes and review of concepts. Without that commitment, the course would be a waste of time. As other reviewers have written, this course requires fundamental knowledge of algebra, trigonometry and elementary calculus. Don't purchase this course if you do not have these skills!. Finally, Dr. Edwards is the consummate teacher. His presentations are clear; he makes complex concepts understandable by taking small bits of a problem and bringing them together into a whole (Integration!). I am a medical school professor, and I learn continually from other professors like Dr. Edwards. He has deep knowledge of his subject, speaks well, has poise, and presents himself with elegant dress and appearance. Dr. Edwards is a role model for students and for teachers.
Date published: 2016-03-04
Rated 5 out of 5 by from A Class Act Professor Edwards is one of the great teachers of my autodidact career: a sharp intellect; a sharp sense of humor; and yes, a sharp dresser. In short, a class act. I've been lucky to have had a few teachers in various disciplines who successfully balanced tough demands on students with elegant presentation of material, but never (until The Great Courses) in mathematics. Perhaps like many otherwise strong students in middle school and high school, I had my early enthusiasm for mathematics quelled to the point of boredom by constant exposure to the New Math as taught in a program with the acronym "SSMCIS" (which students and teachers pronounced as "SMIX"). By the time we reached high school and were studying calculus, even the teachers quietly despised the SSMCIS textbooks — which began the study of calculus with a detailed and perplexing account of continuity — and after some equally quiet rioting, swapped the New Math books for the old Thomas calculus text instead. For many decades after high school, I pondered the problem of re-treading my knowledge of mathematics — not for any practical purpose but for the sake of grasping its essentials — and only recently found a satisfying solution in The Great Courses mathematics series. Kudos, again, to the brilliant Bruce Edwards, as well as to the ever patient and methodical James Sellers, the impressive Edward Burger and Arthur Benjamin, and the engaging Michael Starbird.
Date published: 2016-01-22
Rated 5 out of 5 by from Excellent Comprehensive Calculus III Course I bought this course as a supplement for a Calc III course that I was going to take online. It ended up being a lot more than a supplement. It basically replaced it. I just looked at the lecture topic for the week and popped in the right DVD. If your school offers credit by exam for Calc III, then it is entirely realistic that you could complete this course over a summer and test out in the fall. If you are considering Calc III, this probably goes without saying. But this is certainly not a "from scratch" math class. You could probably keep up after a first semester Calculus class, but you really should have the full single variable curriculum under your belt before you dive into this. Long story short, this course covers everything that you need for college Calc III course, either as a supplement or a full replacement, and it covers it very well.
Date published: 2016-01-03
Rated 5 out of 5 by from Can't recommend any more!!:) Thanks great courses and Prof. Edward for this awesome material. I developed some great concepts from this course. I will be coming back to purchase more. Thanks again.
Date published: 2015-06-09
Rated 5 out of 5 by from Calculus III challenge The college I attended when I took calculus used Larson, Edwards, and Hostetler text. You will never find a better text for Calculus I, II, or III. This video follows the text very closely. If you did not have a good classroom instructor, want to review calculus since you have been away for a while, or if you are curious what calculus is about and whether you should try to take a class, this video was made for you. This is an advanced course. So If you had trouble with Calc I and II, you might want to hold off.
Date published: 2015-03-07
Rated 5 out of 5 by from Another winner This course is closer to a true college course than any other I have seen here. As long as you are comfortable with vectors, derivatives and integrals it is quite doable, especially with such an accomplished teacher. Having had a similar course 49 years ago(!), ostensibly there was little new. However, in my analytical career, only 20% of what is covered here was actually used. (After all, how many times do you need the equation of a plane?) Having said that, it was thoroughly enjoyable to be reacquainted with some wonderful mathematical ideas! Mathematicians often use the word "elegant" to describe the concepts they love so much, and the coverage within this Dr. Edwards course fits that adjective. On a practical note, the area of a region using Green's theorem became a computer program I wrote to evaluate the square footage of my home. [The appraiser shorted me by 100 square ft]. Oh well, I suspect he never learned Green's theorem......
Date published: 2015-02-21
Rated 5 out of 5 by from An Excellent Course and Review Having recently taken and done well in multivariable calculus, I was very happy to find an excellent and detailed review of the course material to help reinforce and to remember important concepts, theories, and formulas. We all have a tendency to forget and this course will bridge that gap for anyone searching for that review, learning about multivariable calculus for the first time, and/or wanting to study advanced math for sheer enjoyment and knowledge.
Date published: 2015-01-25
Rated 5 out of 5 by from Genius educator. Fantastic course! I had thought about purchasing this course for over a year but I thought the price was a bit steep, and the content might be over my head since I have not taken a calculus course in many years. So I bought the course when it went on sale. Now that I have gone thru it and seen what a fantastic course it is, I feel guilty for not paying the list price for it! The success of the course is due to the overwhelming enthusiasm of Professor Edwards which never lags throughout the difficult material. He has organized the material so well that learning from him is always a pleasure. At times his enthusiasm borders on corny, but he always comes across as sincere. As far as the material itself, it requires a lot of starting, stopping, rewinding, reviewing, but that's the way I learn a subject of this difficulty. Professor Edwards presents the material with enough depth to give a good feel for where the equations come from, but he doesn't disrupt the flow of the course by deriving and proving every concept. This makes the course flow without bogging down and holds my attention. To paraphrase Einstein, he makes the course as simple as possible but no simpler. In the end, I'd have to say this is just about the best course I have taken from TGC and it has given me a real sense of accomplishment.
Date published: 2015-01-09
Rated 5 out of 5 by from Very clear presentation This course manages to cover a great deal of material in a very digestible way. An understanding of basic calculus is needed, but other necessary topics such as vector algebra are covered within the material. Very good use is made of diagrams and other visualizations to make complex concepts clear and easy to digest. The workbook is concise but covers the key learning points and provides plenty of practice problems with answers, and as such will function as a very handy textbook. The emphasis is on concepts, developing good intuition, and practical applications rather than on mathematically rigorous proofs. Highly recommended, I needed a refresher on some of the advanced topics after a 40-year hiatus, and got more than I expected - a good understanding of concepts that I had either never learned or never properly understood. The superiority of good video lectures over "talk and chalk" shows through; the professor's presentation stye is nicely paced and easy to understand, and if he ever goes too fast, you can just hit the pause button and reflect on the material.
Date published: 2015-01-08
Rated 5 out of 5 by from One of the better mathematics courses. This is one of the better courses in the science/mathematics section. The professor presents the material very well with abundant examples and often steps back for quick reviews of points from previous lectures or more basic material. His method of "stepping things up a notch" is particularly effective wherein he starts with a simple example and then moves up to more complex 3D. I would like to see a sequel to this course, and several others in the mathematics/science section, 'stepped up a notch or two'. I believe there is a market for more advanced courses.
Date published: 2014-09-28
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Understanding Multivariable Calculus: Problems, Solutions, and Tips
Course Trailer
A Visual Introduction to 3-D Calculus
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.

34 min
Functions of Several Variables
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.

30 min
Limits, Continuity, and Partial Derivatives
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.

30 min
Partial Derivatives-One Variable at a Time
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."

30 min
Total Differentials and Chain Rules
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.

31 min
Extrema of Functions of Two Variables
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.

31 min
Applications to Optimization Problems
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.

31 min
Linear Models and Least Squares Regression
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.

31 min
Vectors and the Dot Product in Space
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.

30 min
The Cross Product of Two Vectors in Space
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.

29 min
Lines and Planes in Space
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.

32 min
Curved Surfaces in Space
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.

31 min
Vector-Valued Functions in Space
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.

31 min
Kepler's Laws-The Calculus of Orbits
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.

30 min
Directional Derivatives and Gradients
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.

30 min
Tangent Planes and Normal Vectors to a Surface
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.

29 min
Lagrange Multipliers-Constrained Optimization
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.

31 min
Applications of Lagrange Multipliers
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.

30 min
Iterated integrals and Area in the Plane
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.

30 min
Double Integrals and Volume
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.

30 min
Double Integrals in Polar Coordinates
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.

31 min
Centers of Mass for Variable Density
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.

30 min
Surface Area of a Solid
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.

31 min
Triple Integrals and Applications
24: Triple Integrals and Applications

Apply your skills in evaluating double integrals to take the next step: triple integrals, which can be used to find the volume of a solid in space. Next, extrapolate the density of planar lamina to volumes defined by triple integrals, evaluating density in its more familiar form of mass per unit of volume.

29 min
Triple Integrals in Cylindrical Coordinates
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.

31 min
Triple Integrals in Spherical Coordinates
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.

30 min
Vector Fields-Velocity, Gravity, Electricity
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.

30 min
Curl, Divergence, Line Integrals
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.

31 min
More Line Integrals and Work by a Force Field
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.

31 min
Fundamental Theorem of Line Integrals
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.

31 min
Green's Theorem-Boundaries and Regions
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.

31 min
Applications of Green's Theorem
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.

30 min
Parametric Surfaces in Space
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.

32 min
Surface Integrals and Flux Integrals
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.

31 min
Divergence Theorem-Boundaries and Solids
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.

29 min
Stokes's Theorem and Maxwell's Equations
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.

34 min
Bruce H. Edwards

I love mathematics and tried to communicate this passion to others, regardless of their mathematical backgrounds.


Dartmouth College


University of Florida

About Bruce H. Edwards

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 Bogota, Colombia, as a Peace Corps volunteer. Professor Edwards has won many teaching awards at the University of Florida, including Teacher of the Year in the College of Liberal Arts and Sciences, Liberal Arts and Sciences Student Council Teacher of the Year, and the University of Florida Honors Program Teacher of the Year. He was selected by the Office of Alumni Affairs to be the Distinguished Alumni Professor for 1991-1993. Professor Edwards has taught a variety of mathematics courses at the University of Florida, from first-year calculus to graduate-level classes in algebra and numerical analysis. He has been a frequent speaker at research conferences and meetings of the National Council of Teachers of Mathematics. He has also coauthored a wide range of mathematics textbooks with Professor Ron Larson. Their textbooks have been honored with various awards from the Text and Academic Authors Association.

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