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## Episodes

- S1 E1 - A Visual Introduction to 3-D CalculusMay 8, 201434minReview 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 & MathematicsFree trial of The Great Courses Signature Collection or buy
- S1 E2 - Functions of Several VariablesMay 8, 201430minWhat 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.Free trial of The Great Courses Signature Collection or buy
- S1 E3 - Limits, Continuity, and Partial DerivativesMay 8, 201430minApply 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.Free trial of The Great Courses Signature Collection or buy
- S1 E4 - Partial Derivatives - One Variable at a TimeMay 8, 201430minDeep 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."Free trial of The Great Courses Signature Collection or buy
- S1 E5 - Total Differentials and Chain RulesMay 8, 201431minComplete 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.Free trial of The Great Courses Signature Collection or buy
- S1 E6 - Extrema of Functions of Two VariablesMay 8, 201431minThe 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.Free trial of The Great Courses Signature Collection or buy
- S1 E7 - Applications to Optimization ProblemsMay 8, 201431minContinue 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.Free trial of The Great Courses Signature Collection or buy
- S1 E8 - Linear Models and Least Squares RegressionMay 8, 201431minApply 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.Free trial of The Great Courses Signature Collection or buy
- S1 E9 - Vectors and the Dot Product in SpaceMay 8, 201430minBegin 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.Free trial of The Great Courses Signature Collection or buy
- S1 E10 - The Cross Product of Two Vectors in SpaceMay 8, 201429minTake 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.Free trial of The Great Courses Signature Collection or buy
- S1 E11 - Lines and Planes in SpaceMay 8, 201432minTurn 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.Free trial of The Great Courses Signature Collection or buy
- S1 E12 - Curved Surfaces in SpaceMay 8, 201431minBeginning 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.Free trial of The Great Courses Signature Collection or buy
- S1 E13 - Vector-Valued Functions in SpaceMay 8, 201431minConsolidate 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.Free trial of The Great Courses Signature Collection or buy
- S1 E14 - Kepler's Laws - The Calculus of OrbitsMay 8, 201430minBlast 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.Free trial of The Great Courses Signature Collection or buy
- S1 E15 - Directional Derivatives and GradientsMay 8, 201430minContinue 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.Free trial of The Great Courses Signature Collection or buy
- S1 E16 - Tangent Planes and Normal Vectors to a SurfaceMay 8, 201429minUtilize 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.Free trial of The Great Courses Signature Collection or buy
- S1 E17 - Lagrange Multipliers - Constrained OptimizationMay 8, 201431minIt'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.Free trial of The Great Courses Signature Collection or buy
- S1 E18 - Applications of Lagrange MultipliersMay 8, 201430minHow 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.Free trial of The Great Courses Signature Collection or buy
- S1 E19 - Iterated integrals and Area in the PlaneMay 8, 201430minWith 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.Free trial of The Great Courses Signature Collection or buy
- S1 E20 - Double Integrals and VolumeMay 8, 201430minIn 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.Free trial of The Great Courses Signature Collection or buy
- S1 E21 - Double Integrals in Polar CoordinatesMay 8, 201431minTransform 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.Free trial of The Great Courses Signature Collection or buy
- S1 E22 - Centers of Mass for Variable DensityMay 8, 201430minWith 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.Free trial of The Great Courses Signature Collection or buy
- S1 E23 - Surface Area of a SolidMay 8, 201431minBring 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.Free trial of The Great Courses Signature Collection or buy
- S1 E24 - Triple Integrals and ApplicationsMay 8, 201429minApply 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.Free trial of The Great Courses Signature Collection or buy
- S1 E25 - Triple Integrals in Cylindrical CoordinatesMay 8, 201431minJust 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.Free trial of The Great Courses Signature Collection or buy

## Extras

### Bonus

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.

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