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Understanding Multivariable Calculus: Problems, Solutions, and Tips

Season 1
Understanding Multivariable Calculus: Problems, Solutions, and Tips, taught by award-winning Professor Bruce H. Edwards, is the next step for students and professionals to expand their knowledge for work or study in many quantitative fields, as well as an intellectual exercise for teachers, retired professionals, and anyone else who wants to understand the amazing applications of 3-D calculus.
201436 episodesTV-PG
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Episodes

  1. S1 E1 - A Visual Introduction to 3-D Calculus
    May 8, 2014
    34min
    TV-PG
    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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  2. S1 E2 - Functions of Several Variables
    May 8, 2014
    30min
    TV-PG
    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.
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  3. S1 E3 - Limits, Continuity, and Partial Derivatives
    May 8, 2014
    30min
    TV-PG
    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.
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  4. S1 E4 - Partial Derivatives - One Variable at a Time
    May 8, 2014
    30min
    TV-PG
    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."
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  5. S1 E5 - Total Differentials and Chain Rules
    May 8, 2014
    31min
    TV-PG
    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.
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  6. S1 E6 - Extrema of Functions of Two Variables
    May 8, 2014
    31min
    TV-PG
    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.
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  7. S1 E7 - Applications to Optimization Problems
    May 8, 2014
    31min
    TV-PG
    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.
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  8. S1 E8 - Linear Models and Least Squares Regression
    May 8, 2014
    31min
    TV-PG
    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.
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  9. S1 E9 - Vectors and the Dot Product in Space
    May 8, 2014
    30min
    TV-PG
    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.
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  10. S1 E10 - The Cross Product of Two Vectors in Space
    May 8, 2014
    29min
    ALL
    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.
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  11. S1 E11 - Lines and Planes in Space
    May 8, 2014
    32min
    TV-PG
    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.
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  12. S1 E12 - Curved Surfaces in Space
    May 8, 2014
    31min
    TV-PG
    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.
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  13. S1 E13 - Vector-Valued Functions in Space
    May 8, 2014
    31min
    TV-PG
    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.
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  14. S1 E14 - Kepler's Laws - The Calculus of Orbits
    May 8, 2014
    30min
    TV-PG
    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.
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  15. S1 E15 - Directional Derivatives and Gradients
    May 8, 2014
    30min
    TV-PG
    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.
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  16. S1 E16 - Tangent Planes and Normal Vectors to a Surface
    May 8, 2014
    29min
    TV-PG
    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.
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  17. S1 E17 - Lagrange Multipliers - Constrained Optimization
    May 8, 2014
    31min
    TV-PG
    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.
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  18. S1 E18 - Applications of Lagrange Multipliers
    May 8, 2014
    30min
    TV-PG
    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.
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  19. S1 E19 - Iterated integrals and Area in the Plane
    May 8, 2014
    30min
    TV-PG
    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.
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  20. S1 E20 - Double Integrals and Volume
    May 8, 2014
    30min
    TV-PG
    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.
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  21. S1 E21 - Double Integrals in Polar Coordinates
    May 8, 2014
    31min
    TV-PG
    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.
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  22. S1 E22 - Centers of Mass for Variable Density
    May 8, 2014
    30min
    TV-PG
    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.
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  23. S1 E23 - Surface Area of a Solid
    May 8, 2014
    31min
    TV-PG
    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.
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  24. S1 E24 - Triple Integrals and Applications
    May 8, 2014
    29min
    TV-PG
    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.
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  25. S1 E25 - Triple Integrals in Cylindrical Coordinates
    May 8, 2014
    31min
    TV-PG
    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.
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Extras

Bonus

Understanding Multivariable Calculus: Problems, Solutions, and Tips
Understanding Multivariable Calculus: Problems, Solutions, and Tips
1minTV-PG
Understanding Multivariable Calculus: Problems, Solutions, and Tips, taught by award-winning Professor Bruce H. Edwards, is the next step for students and professionals to expand their knowledge for work or study in many quantitative fields, as well as an intellectual exercise for teachers, retired professionals, and anyone else who wants to understand the amazing applications of 3-D calculus.
Understanding Multivariable Calculus: Problems, Solutions, and Tips, taught by award-winning Professor Bruce H. Edwards, is the next step for students and professionals to expand their knowledge for work or study in many quantitative fields, as well as an intellectual exercise for teachers, retired professionals, and anyone else who wants to understand the amazing applications of 3-D calculus.
Understanding Multivariable Calculus: Problems, Solutions, and Tips, taught by award-winning Professor Bruce H. Edwards, is the next step for students and professionals to expand their knowledge for work or study in many quantitative fields, as well as an intellectual exercise for teachers, retired professionals, and anyone else who wants to understand the amazing applications of 3-D calculus.

Details

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Producers
The Great Courses
Cast
Bruce H. Edwards
Studio
The Great Courses
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