The course is the second of the two course sequence of calculus of multivariable functions. The first course develops the concepts of derivatives and integrals of functions of several variables, and the basic tools for doing the relevant calculations. This course builds on the foundations of the first course and introduces more advanced topics along with more advanced applications and solved problems. The course is designed with a “content-based” approach, i. e. by solving examples, as many as possible from real life situations.
The “why” and “where“ of the topics are discussed, as much as the “what” and the “how”. The answers to the latter are the “definitions” and “proofs”, while the answers to the first two tell the reason for studying a topic, and the areas where such ideas are used.
The transfer of knowledge through an organized deductive process plays an important role in mathematics (Aristotelian approach). An interactive communication between the teacher and the student through posing questions and answering them leads to an effective method (Socratian method). The design of this course will benefit from the latter whenever feasible.
Why do we study derivatives and integrals? Because derivatives express change, and integrals define the cumulative results of many inputs. Change and growth through time or space are two basic aspects of life. Change is expressed with the difference between two situations, and the cumulative result of many inputs is an additive process. Thus basically, calculus is an extension of what we all learn as early as first grade as addition and subtraction. Calculus enables us to define and calculate instantaneous changes and growth by continuously varying inputs. Instantaneity of the changes and variability of the inputs are handled by infinitesimal quantities. The final results are obtained in the limit where the infinitesimal changes become zero. The limit is the central concept of calculus.
A function defines the relationship between the inputs, which are the independent variables, and outputs which are the dependent variables. The ratio of the infinitesimal changes in the dependent variable to those of the independent variable leads to the concept of the “derivative”. Similarly, the cumulative outputs of entities such as matter, energy, area, surface, volume, etc. are calculated by the sum of the dependent variable weighted by the changes in the independent variable. This operation leads to the concept of “integral”. Just like in Grade One, where we observed that addition and subtraction are the inverses of each other, so are integral and derivative. This complementarity between the derivative and integral is expressed by the two “fundamental theorems of calculus”. All this is studied in the “Calculus of Single Variable Functions”.
Why multivariables? Because real life problems involve several variables. Our environment is defined by three space variables and phenomena evolve in terms of a fourth which is time. People- made phenomena require many more variables. The course offered here is built on the knowledge of calculus of single variable functions and extends the concepts and techniques to multivariable functions. The concepts and techniques are, in most cases, natural extensions and generalizations from those in single variable functions. Hence, each topic will start the review of the fundamental concepts and calculation techniques from the calculus of one variable functions. This review is an opportunity to supplement what a student missed in the earlier course on single variables, while advancing into relevant problems from real life that involve more than one variable.
Recommended Background:
Basic knowledge of the algebra and graphs of one variable polynomial, exponential, logarithmic and trigonometric functions. Basic knowledge of the concepts of derivatives, integrals, the fundamental theorems of calculus, elementary techniques of evaluating derivatives and integrals. The knowledge and understanding of the basic definitions of partial derivatives and double integrals and some basic calculation tools are necessary. The first course of this sequence “Multivariable Calculus I: Fundamental Concepts and Basic Techniques” is not a prerequisite but is recommended.