Understanding Multivariable Calculus: Problems, Solutions, and Tips
Where to Watch Understanding Multivariable Calculus: Problems, Solutions, and Tips
36
Stokes's Theorem and Maxwell's Equations
2014-05-09
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.
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35
Divergence Theorem - Boundaries and Solids
2014-05-09
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.
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34
Surface Integrals and Flux Integrals
2014-05-09
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.
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33
Parametric Surfaces in Space
2014-05-09
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.
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32
Applications of Green's Theorem
2014-05-09
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.
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31
Green's Theorem - Boundaries and Regions
2014-05-09
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.
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30
Fundamental Theorem of Line Integrals
2014-05-09
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.
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29
More Line Integrals and Work by a Force Field
2014-05-09
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.
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28
Curl, Divergence, Line Integrals
2014-05-09
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.
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27
Vector Fields - Velocity, Gravity, Electricity
2014-05-09
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.
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26
Triple Integrals in Spherical Coordinates
1970-01-01
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.
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25
Triple Integrals in Cylindrical Coordinates
2014-05-09
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,?
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24
Triple Integrals and Applications
2014-05-09
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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23
Surface Area of a Solid
2014-05-09
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.
Watch Understanding Multivariable Calculus: Problems, Solutions, and Tips Season 1 Episode 23 Now
22
Centers of Mass for Variable Density
2014-05-09
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.
Watch Understanding Multivariable Calculus: Problems, Solutions, and Tips Season 1 Episode 22 Now
21
Double Integrals in Polar Coordinates
2014-05-09
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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20
Double Integrals and Volume
2014-05-09
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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19
Iterated integrals and Area in the Plane
2014-05-09
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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18
Applications of Lagrange Multipliers
2014-05-09
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.
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17
Lagrange Multipliers - Constrained Optimization
2014-05-09
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.
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16
Tangent Planes and Normal Vectors to a Surface
2014-05-09
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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15
Directional Derivatives and Gradients
2014-05-09
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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14
Kepler's Laws - The Calculus of Orbits
2014-05-09
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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13
Vector-Valued Functions in Space
2014-05-09
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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12
Curved Surfaces in Space
2014-05-09
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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11
Lines and Planes in Space
2014-05-09
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.
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10
The Cross Product of Two Vectors in Space
2014-05-09
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.
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9
Vectors and the Dot Product in Space
2014-05-09
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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8
Linear Models and Least Squares Regression
2014-05-09
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.
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7
Applications to Optimization Problems
2014-05-09
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.
Watch Understanding Multivariable Calculus: Problems, Solutions, and Tips Season 1 Episode 7 Now
6
Extrema of Functions of Two Variables
2014-05-09
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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5
Total Differentials and Chain Rules
2014-05-09
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 ?
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4
Partial Derivatives - One Variable at a Time
2014-05-09
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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3
Limits, Continuity, and Partial Derivatives
2014-05-09
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.
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2
Functions of Several Variables
2014-05-09
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.
Watch Understanding Multivariable Calculus: Problems, Solutions, and Tips Season 1 Episode 2 Now
1
A Visual Introduction to 3-D Calculus
2014-05-09
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.
Watch Understanding Multivariable Calculus: Problems, Solutions, and Tips Season 1 Episode 1 Now
Understanding Multivariable Calculus: Problems, Solutions, and Tips is a series categorized as a new series. Spanning 1 seasons with a total of 36 episodes, the show debuted on 2014. The series has earned a no reviews from both critics and viewers. The IMDb score stands at undefined.
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How can I watch Understanding Multivariable Calculus: Problems, Solutions, and Tips online? Understanding Multivariable Calculus: Problems, Solutions, and Tips is available on The Great Courses Signature Collection with seasons and full episodes. You can also watch Understanding Multivariable Calculus: Problems, Solutions, and Tips on demand at Amazon Prime, Amazon online.
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