Calculus III (Multivariable Calculus)
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This course involves a study of functions of two or more variables using the principles of calculus, vector analysis, and parametric equations. This is the third course of the calculus sequence required of engineering, physics, and mathematics majors. This course contains a series of video tutorials that are broken up in various levels. Each video builds upon the previous one.
This course contains a series of video tutorials that are broken up into various levels. Each video builds upon the previous one. Level I videos lay out the theoretical frame work to successfully tackle on problems covered in the next videos.
These videos can be used as a stand along course or as a supplement to your current Calculus III class.
This course is consistently being populated with new videos.
This course is for anyone who wants to fortify their understanding of calculus III or anyone that wishes to learn calculus III can benefit from this course.
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Section 1: Introduction | |||
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Lecture 1 | 01:01 | ||
This video gives an overview of what this course will cover. |
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Section 2: Three Dimensional Coordinate Systems | |||
Lecture 2 | 11:48 | ||
This video goes over the basic concepts and terminology of one dimensional, and two dimensional coordinate systems. This video concludes with an introduction to three dimensional coordinate systems as a starting point to successfully study multivariable calculus. |
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Lecture 3 | 10:21 | ||
This video goes over basic equations of a two dimensional coordinate system are presented to illustrate the similarities to a three dimensional coordinate system. In addition, the equations of the coordinate planes are also discussed. This video concludes with an introduction to projections in three dimensional space. |
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Lecture 4 | 10:30 | ||
This video goes over 5 examples covering the proper way to graph equations in R cubed. Equations covered include planes, cylinder, and parabolic cylinder. |
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Lecture 5 | 10:52 | ||
This video goes over common formulas used in a three dimensional coordinate system, this video covers the midpoint formula and a derivation of the distance formula in three dimensions is also presented. |
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Lecture 6 | 10:47 | ||
This video covers two basic examples requiring the use of the distance formula in three dimensions. In addition, the equation of a sphere is also derived. |
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Lecture 7 | 10:26 | ||
This video covers three slightly more challenging examples requiring the use of the distance formula in three dimensions. |
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Lecture 8 | 06:49 | ||
This video covers three basic examples that requires the use of the equation of a sphere. |
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Lecture 9 | 09:55 | ||
This video covers three intermediate examples that requires the use of the equation of a sphere. Two examples illustrate how to find the center and radius of a sphere by rewriting an equation into its standard form by completing the square. The final example illustrates how to find an equation of a sphere that is tangent to each of the coordinate planes. |
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Lecture 10 | 08:12 | ||
This video covers two challenging examples that requires the use of the equation of a sphere. The first example involves finding the an equation of a sphere that is constrained in the first octant. The second example involves finding an equation of a sphere by solving a system of equations. |
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Lecture 11 | 11:36 | ||
This video covers 2 examples illustrating the appropriate way of graphing equations in three dimensional space that are restricted to a given interval. In addition, this video goes over 5 examples illustrating how to graph inequalities in three dimensional space. |
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Section 3: Two Dimensional Vectors | |||
Lecture 12 | 08:08 | ||
This video is a review of Two Dimensional Vectors. This video goes over the basic concepts and terminology of vectors in a plane. Topics include: Vectors, Magnitude of a Vector, Equivalent Vectors, and Vector Notation. This video also goes over 2 examples. |
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Lecture 13 | 10:09 | ||
This video is a review of Two Dimensional Vectors. Topics include vectors in a coordinate system, vectors in standard position, and component form of a vector. |
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Lecture 14 | 09:33 | ||
This video is a review of Two Dimensional Vectors. This video goes over 5 examples covering how to write the component form of a vector and sketching vectors in standard position. |
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Lecture 15 | 11:03 | ||
This video is a review of Two Dimensional Vectors. This video goes vector operations also known as vector arithmetic. Topics include: geometric interpretation of scalar multiplication of a vector, vector addition, and vector subtraction. |
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Lecture 16 | 10:07 | ||
This video is a review of Two Dimensional Vectors. This video goes over 12 examples covering vector addition, vector subtraction and scalar multiplication. These problems are solved by using the geometric interpretation of these particular vector operations. |
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Lecture 17 | 11:30 | ||
This video is a review of Two Dimensional Vectors. This video goes over how to algebraically find the scalar multiple of a vector and vector addition. The concept of parallel vector is also introduced along with 2 examples. |
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Lecture 18 | 10:40 | ||
This video is a review of Two Dimensional Vectors. This video goes over 5 examples that make use of the algebraic definition of scalar multiplication and vector addition. |
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Lecture 19 | 09:57 | ||
This video is a review of Two Dimensional Vectors. This video goes over properties of vector operations. Properties are also proven geometrically and algebraically. |
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Lecture 20 | 10:29 | ||
This video is a review of Two Dimensional Vectors. This video goes over unit vectors, standard unit vectors and direction of vectors. |
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Lecture 21 | 10:30 | ||
This video is a review of Two Dimensional Vectors. This video goes over 6 examples that make use of unit vectors. |
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Lecture 22 | 09:08 | ||
This video is a review of Two Dimensional Vectors. This video goes over applications of vectors. 3 examples are covered illustrating how to find the resultant force. |
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Lecture 23 | 10:56 | ||
This video is a review of Two Dimensional Vectors. This video goes over applications of vectors. 3 examples are covered illustrating how to solve static equilibrium problems. |
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Lecture 24 | 11:24 | ||
This video is a review of Two Dimensional Vectors. This video goes over applications of vectors. 3 examples are covered illustrating how to solve distance and bearing problems. |
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Section 4: Three Dimensional Vectors | |||
Lecture 25 | 09:20 | ||
This video covers Three Dimensional Vectors. This video goes over the various properties associated with three dimensional vectors. 3 basic examples are also covered illustrating how to solve problems that make use of vectors in space. |
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Lecture 26 | 09:15 | ||
This video covers Three Dimensional Vectors. This video goes over the various properties associated with three dimensional vectors. 6 intermediate examples are covered illustrating how to solve problems that make use of vectors in space. |
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Lecture 27 | 09:52 | ||
This video covers Three Dimensional Vectors. This video goes over the various properties associated with three dimensional vectors. 4 intermediate examples including a static equilibrium problem are covered. |
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Section 5: The Dot Product | |||
Lecture 28 | 11:22 | ||
This video goes over the dot product also known as the scalar product. This video covers the geometric interpretation of the dot product by going over 5 distinct cases where the angle between the vectors varies. |
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Lecture 29 | 08:13 | ||
This video goes over the dot product also known as the scalar product. In this video we will derive another method to compute the dot product between two vectors by using their components. We will also cover the properties of the dot product. |
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Lecture 30 | 08:31 | ||
This video goes over the dot product also known as the scalar product. This video goes over 11 examples illustrating how to solve problems that make use of the geometric and component definition of the dot product. |
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Lecture 31 | 09:00 | ||
This video goes over the dot product also known as the scalar product. This video goes over 5 examples illustrating how to solve problems that make use of the geometric and component definition of the dot product. | |||
Lecture 32 | 10:19 | ||
This video goes over the dot product also known as the scalar product. This video goes over a proof for the geometric definition of the dot product. This video also goes over 4 examples illustrating how to find the angle between two vectors. |
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Lecture 33 | 11:17 | ||
This video goes over the dot product also known as the scalar product. This video goes over 5 Intermediate level examples that require the use of dot product. |
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Lecture 34 | 10:29 | ||
This video goes over the dot product also known as the scalar product. This video goes over 3 challenging examples that require the use of dot product. |
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Lecture 35 | 08:11 | ||
This video goes over the dot product also known as the scalar product. This video covers an application of the dot product specifically, the scalar projection, vector projection, and orthogonal projection. |
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Lecture 36 | 06:04 | ||
This video goes over the dot product also known as the scalar product. This video goes over 3 examples illustrating how to find the scalar projection, vector projection, and orthogonal projection. |
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Lecture 37 | 08:12 | ||
This video goes over the dot product also known as the scalar product. This video covers direction angles and direction cosines. 3 examples are also covered illustrating how to solve problems involving direction angles and direction cosines. |
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Lecture 38 | 10:07 | ||
This video goes over the dot product also known as the scalar product. This video goes over how to find the work done by a constant force. 4 examples are also presented illustrating how to solve work problems. |
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Lecture 39 | 07:59 | ||
This video goes over the dot product also known as the scalar product. This video ends the dot product series by going over 3 proofs specifically the Cauchy-Schwarz Inequality, Triangle Inequality and the Parallelogram Law. |
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Section 6: The Cross Product | |||
Lecture 40 | 09:21 | ||
This video introduces the third way of multiplying vectors called the cross product also known as the vector product and sometimes refereed to as the area product. This video will cover the geometric definition of the cross product. |
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Lecture 41 | 12:33 | ||
This video introduces the third way of multiplying vectors called the cross product also known as the vector product and sometimes refereed to as the area product. This video will cover the component definition of the cross product. A review of determinants is also presented. |
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Lecture 42 | 10:16 | ||
This video goes over 5 examples illustrating how to find the cross product of two vectors in space by using both the geometric and component definition of the cross product. |
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Lecture 43 | 09:04 | ||
This video goes over 8 examples illustrating how to find the cross product of two vectors in space by using both the geometric and component definition of the cross product. |
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Lecture 44 | 12:33 | ||
This video goes over various algebraic properties of the cross product. Proofs of these properties are also presented as well as 4 examples. |
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Lecture 45 | 11:28 | ||
This video goes over various geometric properties of the cross product. Proofs of these properties are also presented as well as 2 examples. |
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Lecture 46 | 09:38 | ||
This video goes over the scalar triple product also known as the triple scalar product and its use in finding the volume of a parallelepiped and determining of 3 vectors are coplanar. |
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Lecture 47 | 09:27 | ||
This video goes over a second application of the cross product and covers the basic concepts of torque which is also known as moment. |
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Lecture 48 | 09:12 | ||
This video goes over 3 torque examples. This video also ends the cross product series. |
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Section 7: Conclusion | |||
Lecture 49 | 00:45 | ||
This video gives an overview of what was taught in this course. |
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