Watch Understanding Multivariable Calculus: Problems, Solutions, and Tips

Name: Understanding Multivariable Calculus: Problems, Solutions, and Tips
Genre: Documentary & Biography
Released: 2014-05-09
Cast: Bruce H. Edwards

2014
1 Season

Understanding Multivariable Calculus: Problems, Solutions, and Tips from The Great Courses Signature Collection starring Bruce H. Edwards is an enlightening series designed to assist learners to dive deeper into the complex world of multivariable calculus. Unlike introductory calculus, which primarily emphasizes one-dimensional functions, multivariable calculus focuses on functions of several variables, such as three-dimensional space.

As a professor of mathematics at the University of Florida, Bruce H. Edwards brings his expertise to the show by breaking down the often perplexing expertise of multivariable calculus into manageable and understandable concepts. His style of teaching engages learners in the learning process, using techniques such as highlighting key points and correcting common errors.

The show's sixteen lectures are arranged sequentially to help learners master multivariable calculus progressively. The first lecture starts with the fundamentals of vector calculus and matrix algebra, followed by calculus topics like partial derivatives, optimization, Taylor series, and multiple integrals. Instructors use a combination of worked-out problems, practical examples, and streamlined graphics to help students comprehend these mathematical concepts.

The series employs real-world applications to make calculus more understandable, practical, and therefore more fascinating to learners. For example, one of the lectures explains how to use multivariable calculus to calculate the deflection of a beam under pressure or how to find the mass of a complex object with a varying density.

What makes Understanding Multivariable Calculus: Problems, Solutions, and Tips stand out is the interactivity provided. With each lecture, students receive a workbook that guides them through the presented topics, allowing them to solve problems along with the instructor. This hands-on feature makes it easier for students to comprehend complex concepts and allows them to engage in discussions and think critically.

In addition to the workbook, students have access to a course webpage, which provides additional resources such as problem sets and quizzes, to help reinforce the topics presented in the lectures. The course page also allows students to connect with their instructor and other classmates, providing a dynamic learning experience.

Bruce H. Edwards is a skilled lecturer, and his teaching style keeps learners engaged throughout the course. He uses appropriate humor and practical examples to bring students into the mathematical world, making them feel more comfortable with the subject. The enthusiasm and expertise he possesses are undoubtedly a significant factor in making this series one of the most impactful and successful multivariable calculus courses available.

Overall, Understanding Multivariable Calculus: Problems, Solutions, and Tips from The Great Courses Signature Collection starring Bruce H. Edwards is a well-organized and apt course for learners of various backgrounds. Whether you are a beginner or an experienced student looking for a refresher course, this series provides an excellent opportunity to learn multivariable calculus from one of the best teachers. The series offers a blend of practical examples and mathematical concepts to help learners gain a comprehensive understanding of the subject. The optional exercises and interactive features make it an optimal choice for anyone looking to learn multivariable calculus in depth.

Understanding Multivariable Calculus: Problems, Solutions, and Tips is a series that ran for 1 seasons (36 episodes) between May 9, 2014 and on The Great Courses Signature Collection

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Seasons

36. Stokes's Theorem and Maxwell's Equations

May 9, 2014

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.

35. Divergence Theorem - Boundaries and Solids

May 9, 2014

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.

34. Surface Integrals and Flux Integrals

May 9, 2014

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.

33. Parametric Surfaces in Space

May 9, 2014

In this episode, extend your understanding of surfaces by defining them in terms of parametric equations. Learn to graph parametric surfaces and to calculate surface area.

32. Applications of Green's Theorem

May 9, 2014

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.

31. Green's Theorem - Boundaries and Regions

May 9, 2014

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.

30. Fundamental Theorem of Line Integrals

May 9, 2014

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.

29. More Line Integrals and Work by a Force Field

May 9, 2014

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.

28. Curl, Divergence, Line Integrals

May 9, 2014

Use the gradient vector to find the curl and divergence of a field, both 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.

27. Vector Fields - Velocity, Gravity, Electricity

May 9, 2014

In your introduction to vector fields, 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.

26. Triple Integrals in Spherical Coordinates

January 1, 1970

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.

25. Triple Integrals in Cylindrical Coordinates

May 9, 2014

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.

24. Triple Integrals and Applications

May 9, 2014

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.

23. Surface Area of a Solid

May 9, 2014

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.

22. Centers of Mass for Variable Density

May 9, 2014

With these new methods of evaluating integrals over a region, we can apply these concepts to the realm of physics. Continuing from the previous episode, 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.

21. Double Integrals in Polar Coordinates

May 9, 2014

Transform 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.

20. Double Integrals and Volume

May 9, 2014

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.

19. Iterated integrals and Area in the Plane

May 9, 2014

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.

18. Applications of Lagrange Multipliers

May 9, 2014

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 an earlier episode 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.

17. Lagrange Multipliers - Constrained Optimization

May 9, 2014

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.

16. Tangent Planes and Normal Vectors to a Surface

May 9, 2014

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.

15. Directional Derivatives and Gradients

May 9, 2014

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 episodes.

14. Kepler's Laws - The Calculus of Orbits

May 9, 2014

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.

13. Vector-Valued Functions in Space

May 9, 2014

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.

12. Curved Surfaces in Space

May 9, 2014

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 2-D shapes that lay hidden in new vector equations, and observe surfaces of revolution in three-dimensional space.

11. Lines and Planes in Space

May 9, 2014

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.

10. The Cross Product of Two Vectors in Space

May 9, 2014

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 an earlier episode to define the triple scalar product, and use it to evaluate the volume of a parallelepiped.

9. Vectors and the Dot Product in Space

May 9, 2014

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.

8. Linear Models and Least Squares Regression

May 9, 2014

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.

7. Applications to Optimization Problems

May 9, 2014

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.

6. Extrema of Functions of Two Variables

May 9, 2014

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.

5. Total Differentials and Chain Rules

May 9, 2014

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.

4. Partial Derivatives - One Variable at a Time

May 9, 2014

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."

3. Limits, Continuity, and Partial Derivatives

May 9, 2014

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.

2. Functions of Several Variables

May 9, 2014

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.

1. A Visual Introduction to 3-D Calculus

May 9, 2014

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 curiosities unique to functions of more than one variable. #Science & Mathematics

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Where to Watch Understanding Multivariable Calculus: Problems, Solutions, and Tips

Understanding Multivariable Calculus: Problems, Solutions, and Tips is available for streaming on the The Great Courses Signature Collection website, both individual episodes and full seasons. You can also watch Understanding Multivariable Calculus: Problems, Solutions, and Tips on demand at Amazon Prime and Amazon.