Calculus Revisited Videos Part 2 - Functions of Several Variables
 
The Game of Mathematics      

Block 1: Mathematical Structure and Vectors

Professor Gross introduces the arithmetic of vectors in terms of the structure of a game. In particular, he shows how the arithmetic of vectors helps us define planes and lines analytically.

1) The Game of Mathematics

2) Vectors Revisited - Arrow Arithmetic

3) 3-dimensional Vectors

4) The Dot Product

5) The Cross Product

6) Lines and Planes

Block 2: Functions
In Calculus Revisited Part 1, Professor Gross discussed the calculus of a single real variable in which the domain of a function was a  subset of the real numbers. Geometrically speaking, the domain of a function was a subset of the x-axis. In this block he generalizes the domain as being a subset of either the two-dimensional xy-plane and/or the three-dimensional xyz-space. In the language of vectors, in this block a function maps 2 and 3 dimensional vectors into the set of real numbers. He then uses these functions to show how we compute the velocity and acceleration of an object moving in space.
 

1) Vector Functions of Scalar Variables

2) Tangent and Normal Vectors

3) Polar Coordinates

4) Vectors and Polar Coordinates

Block 3: Real Valued Functions of Several Real Variables
In this block Professor Gross uses the "game of mathematics" concept to develop an analytical way to extend the domain of a function to beyond 3 dimensions. In particular, he shows how by using vector arithmetic, the rules of arithmetic that were used in developing the calculus of a single variable turn out to be the same that we use to develop the calculus of several variables. This leads to a discussion of how we replace the concept of slope in the 2 and 3-dimensional calculus by such concepts as the directional derivative when dealing with more than 3 dimensions.
 

1) n-Dimensional Vector Spaces

2) Introduction to the Calculus of Several Variables

3) The Directional Derivative

4) The Chain Rule

5 Integrals Involving Parameters

6) Exact Differentials

Block 4: Matrix Algebra
Block 4 extends the concept of inverse functions to the case where y = f(x) with y = (y1, y2, ..., yn) and  x = (x1, x2, ..., xn). In more user-friendly terms this block asks us to determine when and how the system of equations that expresses y1, y2, ... and yn as functions of  x1, x2, and xn  can  be  "inverted" to express x1, x2, ... and xn as functions  of  y1, y2, ... and yn. This motivates the study of matrix algebra since the process of inverting an n x n square matrix is used to show how we decide whether a function f(x1, x2, ..., xn) has an inverse and how we find the inverse function if it exists.

1) Linearity Revisited

2) Matrix Algebra

3) Inverting a Square Matrix

4) Inverting General Systems of Equations

5) Max-min Problems in Several Variables

Block 5: Line Integrals along closed curves (Green's Theorem)

1) Double (Multiple) Sums

2) The Fundamental Theorem

3) Multiple Integrals and the Jacobian

4) Line Integrals

5) Green's Theorem

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