Understanding Multivariable Calculus: Problems, Solutions, and Tips
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Episodes
- S1 E1 - A Visual Introduction to 3-D Calculus
May 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 new curiosities unique to functions of more than one variable.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E2 - Functions of Several Variables
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E3 - Limits, Continuity, and Partial Derivatives
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E4 - Partial Derivatives - One Variable at a Time
May 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.”Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E5 - Total Differentials and Chain Rules
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E6 - Extrema of Functions of Two Variables
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E7 - Applications to Optimization Problems
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E8 - Linear Models and Least Squares Regression
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E9 - Vectors and the Dot Product in Space
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E10 - The Cross Product of Two Vectors in Space
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E11 - Lines and Planes in Space
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E12 - Curved Surfaces in Space
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E13 - Vector-Valued Functions in Space
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E14 - Kepler's Laws - The Calculus of Orbits
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E15 - Directional Derivatives and Gradients
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E16 - Tangent Planes and Normal Vectors to a Surface
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E17 - Lagrange Multipliers - Constrained Optimization
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E18 - Applications of Lagrange Multipliers
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E19 - Iterated integrals and Area in the Plane
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E20 - Double Integrals and Volume
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E21 - Double Integrals in Polar Coordinates
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E22 - Centers of Mass for Variable Density
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E23 - Surface Area of a Solid
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E24 - Triple Integrals and Applications
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, or buy - S1 E25 - Triple Integrals in Cylindrical Coordinates
May 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.Subscribe for $1.99/month for 2 month(s) and $7.99/month thereafter, 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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