Lecture 21: The Fundamental Theorem Extended

The video script begins with a review of the course material on calculus, introduced through the example of a car moving along a straight road. The car's position at any given time is represented by a real-valued function of one real variable, which is used to develop the foundational concepts of derivatives and integrals. The instructor emphasizes the extension of these concepts to more complex scenarios, such as motion in a plane and functions of several variables, including partial derivatives and multiple integrals.

The script then delves into the fundamental theorem of calculus, which relates the derivative and the integral, using the analogy of a car's position and velocity functions. The instructor proposes extending this theorem to more complex situations, such as hiking up a mountain slope, which involves functions of two variables. The concept of a topographical map as a function of two variables is introduced, setting the stage for discussing how to find elevation changes, akin to the fundamental theorem's application in one-variable calculus.

The third paragraph discusses the historical method of surveying to determine elevation changes on a mountain before the advent of GPS technology. The process involved incremental measurements of rising and falling over short distances to determine the altitude at various points. This method is likened to an integral, as it accumulates changes in altitude along a path. Two strategies for finding elevation differences are presented: direct subtraction from a topographical map or summing incremental changes along a walked path, both of which are reminiscent of the fundamental theorem of calculus.

This paragraph introduces the concept of partial derivatives in the context of a two-variable function representing terrain elevation. The partial derivatives with respect to x and y indicate the steepness of the terrain in those respective directions. The instructor explains how to calculate the change in altitude when taking a diagonal path by combining the effects of moving in the x and y directions, using the partial derivatives to determine the ascent or descent in each direction.

The script presents the fundamental theorem for line integrals, which is an extension of the fundamental theorem of calculus to deal with changes in altitude along a path in two-dimensional space. The theorem is explained through the integral of the partial derivatives of a function f(x, y) with respect to x and y, multiplied by the respective path components. The integral accumulates the changes in altitude along the path, which is equivalent to the difference in function values at the start and end points of the path, regardless of the path taken.

The sixth paragraph explores the application of calculus in modeling fluid dynamics, specifically the behavior of wind. The wind's direction and speed at each point in space are described using a velocity vector field. The Divergence Theorem, a variation of the fundamental theorem of calculus, is introduced as a means to understand the net flow of gas across a containing surface, relating local expansion or contraction rates to the overall volume flow per unit time.

The final paragraph highlights the broader impact of various theorems that extend the fundamental theorem of calculus, such as Green's theorem, the Divergence Theorem (also known as Gauss's theorem), and Stokes' theorem. These theorems have critical applications in fields like fluid dynamics, electricity, and magnetism. The story of George Green, a self-taught mathematician who contributed significantly to these fields, is shared to illustrate the profound impact that mathematical theories can have on industrial development.