**Understanding Multivariable Calculus: Problems, Solutions, and Tips**

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*Genre: eLearning Video / Calculus, Math*

Calculus offers some of the most astounding advances in all of mathematicsâ€"reaching far beyond the two-dimensional applications learned in first-year calculus. We do not live on a sheet of paper, and in der to understand and solve rich, real-wld problems of me than one variable, we need multivariable calculus, where the full depth and power of calculus is revealed.

Whether calculating the volume of odd-shaped objects, predicting the outcome of a large number of trials in statistics, even predicting the weather, we depend in myriad ways on calculus in three dimensions. Once we grasp the fundamentals of multivariable calculus, we see how these concepts unfold into new laws, entire new fields of physics, and new ways of approaching once-impossible problems.

**With multivariable calculus, we get**

partial derivatives, which are the building blocks of partial differential equations;

new tools f optimization, taking into account as many variables as needed;

vect fields that give us a peek into the wkings of fluids, from hydraulic pistons to ocean currents and the weather;

line integrals that determine the wk done on a path through these vect fields;

new codinate systems that enable us to solve integrals whose solutions in Cartesian codinates may be difficult to wk with;

fce fields, including those of gravitation and electricity; and

mathematical definitions of planes and surfaces in space, from which entire fields of mathematics such as topology and differential geometry arise.

Understanding Multivariable Calculus: Problems, Solutions, and Tips, taught by award-winning Profess Bruce H. Edwards of the University of Flida, brings the concepts of calculus together in a much deeper and me powerful way. Building from an understanding of basic concepts in Calculus I, it is a full-scope course that encompasses all the key topics of multivariable calculus, together with brief reviews of needed concepts as you go along. This course is the next step f students and professionals to expand their knowledge f wk study in mathematics, statistics, science, engineering and to learn new methods to apply to their field of choice. Itâ€™s also an eye-opening intellectual exercise f teachers, retired professionals, and anyone else who wants to understand the amazing applications of 3-D calculus.

Designed f anyone familiar with basic calculus, Understanding Multivariable Calculus follows, but does not essentially require knowledge of, Calculus II. The few topics introduced in Calculus II that do carry over, such as vect calculus, are here briefly reintroduced, but with a new emphasis on three dimensions.

Your main focus throughout, in a series of 36 comprehensive lectures that extend beyond what is typically taught in university classrooms, is on deepening and generalizing fundamental tools of integration and differentiation to functions of me than one variable. Under the expert guidance of Profess Edwards, youâ€™ll embark on an exhilarating journey through the concepts of multivariable calculus, enlivened with real-wld examples and beautiful animated graphics that lift calculus out of the textbook and into our three-dimensional wld.

**Unlock the Full Power of Calculus**

Too frequently, students end their education in higher-level math after a year of fundamental calculusâ€"thereby missing out on this capstone course that makes possible consideration of problems that have the dimensions and complexity of real life. With the tools and techniques of multivariable calculus, students will be able to understand and solve complex problems arising in a wide array of fields in an elegant manner.

Use the gradient vect to optimize a function subject to a constraint using Lagrange multipliers.

Determine wk done on a path with line integrals, and find the flow through a surface with surface integrals.

See how Isaac Newton used multivariable calculus to prove Johannes Keplerâ€™s laws of bits.

Exple the properties of fluids and the relationship of vect fields with path integrals using Stokesâ€™s theem, Greenâ€™s theem, and the Divergence theem.

Understand why Maxwell was able to discover underlying unities between electricity and magnetism that no one had been able to identify befe.

In Understanding Multivariable Calculus, you will begin your journey in familiar territy as you jump into three dimensions with standard Cartesian codinates. Youâ€™ll see the fundamentals you learned in Calculus I extrapolated to three dimensions, as derivatives and the Extreme Value theem are applied with me variables. Then observe how an extra dimension enables partial derivatives to provide new infmation, including how they can be combined into a total differential that enables changes in a multivariable function to be approximated with its partial derivatives.

By adding this extra dimension to vects, you will be given surprising new insight into surfaces and volumes from perspectives you never considered. Youâ€™ll view how vects can be combined with recognizable techniques from geometry and algebra to yield parametric equations that are both powerful and simple in defining lines and planes in space.

Next, youâ€™ll see old integrals in a new light, as two definite integrals with two different variables are computed as an iterated integral. By tweaking this method, youâ€™ll learn to compute double integrals f area, as well as triple integrals f volumeâ€"as well as some helpful techniques involving basic algebra and new codinate systems f setting up your integrals f success.

Finally, witness a truly wonderful thing happen when these new double and triple integrals are combined with what youâ€™ve learned in previous lectures about vects and derivatives: Entire new fields of physics explode into existence. Watch as line integrals, surface integrals, curl, divergence, and flux are derived and illuminate fluid mechanics. Then youâ€™ll see how the famous theems of Green and Stokes and the Divergence theem unite these integrals, and be granted insight into how Maxwell derived his equations that gave birth to the unified study of electromagnetism.

**A New Look at Old Problems**

How do you integrate over a region of the xy plane that canâ€™t be defined by just one standard y = f(x) function? Multivariable calculus is full of hidden surprises, containing the answers to many such questions. In Understanding Multivariable Calculus, Profess Edwards unveils powerful new tools in every lecture to solve old problems in a few steps, turn impossible integrals into simple ones, and yield exact answers where even calculats can only approximate.

And with techniques using new codinate systems, new integrals, and new theems uniting them all, you will be able to integrate volumes and surface areas directly with double and triple integrals;

define easily differentiable parametric equations f a function using vects; and

utilize polar, cylindrical, and spherical codinates to evaluate double and triple integrals whose solutions are difficult in standard Cartesian codinates.

These tools are essential in fields such as statistics, engineering, and physics where equations arise that cannot be wked with easily using the conventional Cartesian codinate system.

Profess Edwards leads you through these new techniques with a clarity and enthusiasm f the subject that make even the most challenging material accessible and enjoyable. From the very first lecture, youâ€™ll see why Profess Edwards has won teaching awards at the University of Flida.

**Discover at Your Own Pace**

With 36 lectures featuring graphics animated with state-of-the-art software that brings three-dimensional surfaces and volumes to life, this course will provide you with a view of multivariable calculus beyond whatâ€™s available in textbooks and lecture halls. Using the accompanying illustrated wkbook, you are free to move at your own pace to grasp the powerful tools of multivariable calculus to your own satisfaction.

This course offers a uniquely self-contained approach, appealing to a wide array of backgrounds and experience levels. Understanding Multivariable Calculus offers students and professionals in virtually every quantitative field as well as anyone who is intrigued about math a chance to better understand the full potential of one of the crowning mathematical achievements of humankind: calculus.

__Lectures:__

01 A Visual Introduction to 3-D Calculus

02 Functions of Several Variables

03 Limits, Continuity, and Partial Derivatives

04 Partial Derivativesâ€"One Variable at a Time

05 Total Differentials and Chain Rules

06 Extrema of Functions of Two Variables

07 Applications to Optimization Problems

08 Linear Models and Least Squares Regression

09 Vects and the Dot Product in Space

10 The Cross Product of Two Vects in Space

11 Lines and Planes in Space

12 Curved Surfaces in Space

13 Vect-Valued Functions in Space

14 Keplerâ€™s Lawsâ€"The Calculus of bits

15 Directional Derivatives and Gradients

16 Tangent Planes and Nmal Vects to a Surface

17 Lagrange Multipliersâ€"Constrained Optimization

18 Applications of Lagrange Multipliers

19 Iterated integrals and Area in the Plane

20 Double Integrals and Volume

21 Double Integrals in Polar Codinates

22 Centers of Mass f Variable Density

23 Surface Area of a Solid

24 Triple Integrals and Applications

25 Triple Integrals in Cylindrical Codinates

26 Triple Integrals in Spherical Codinates

27 Vect Fieldsâ€"Velocity, Gravity, Electricity

28 Curl, Divergence, Line Integrals

29 Me Line Integrals and Wk by a Fce Field

30 Fundamental Theem of Line Integrals

31 Greenâ€™s Theemâ€"Boundaries and Regions

32 Applications of Greenâ€™s Theem

33 Parametric Surfaces in Space

34 Surface Integrals and Flux Integrals

35 Divergence Theemâ€"Boundaries and Solids

36 Stokesâ€™s Theem and Maxwell's Equations

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