Modern Introduction to Differentials

(not in the Stewart's book, helps to handle multiple integrals and vector calculus concepts; the Lecturer will gradually be introducing everything we need; those interested more might check with the well-known Calculus by Kaplan. The Stewart's presents an extremely simplified version of differentials. I will design all test problems in such a way that technically a student will be able to get up to "A" on the basis of the Stewart's book alone). It is based on the important notion of DUALITY. A dual to a vector space V is the totality W of all linear functions on V. Of course, W, itself, is a vector space, since the sum f + g of two linear functions is a linear function; and, if s is a scalar, sf is a linear function. W contains a zero element (which assignes a number zero to each vector v from V). All usual properties of vector spaces take place for W. Always, they are of one and the same dimension: dim W = dim V. Consider the x-axis as a one-dimensional vector space V, "i" is its basic vector. By "dx" mathematicians denote such a basic element of the dual space W, that the value of dx on i is the number 1. Since dx is a linear function, this determines dx completely. "dx" is defined everywhere on V. Let V be a three-dimensional vector space with a standard basis i, j, k. By dx, dy, dz we denote the "dual basis" in W. Namely, dx maps i into 1, j into 0, k into 0; similarly, dy maps i and k into 0, dy maps j into 1; dz maps i, j into zeros, dz assigns 1 to k. Having the linearity in mind, the above defines dx, dy, dz as real-valued functions on V. Any other linear function on V is a unique linear combination of dx, dy, dz. Multiple integrals will be introduced as integrals of a one-form (dim=1), of a two-form (double integral), etc. The "d" can now be interpreted as an OPERATOR of DIFFERENTIATION. It maps functions into one-forms, one-forms into two-forms, etc. All usual rules apply (linearity, the product rule, etc.). Now, dx can also be interpreted as a result of application of an operator "d" to a coordinate function "x" (depending on the context, "x" is a function of a single variable, of two variables (x,y), of three variables (x,y,z), etc.).

The gradient of a function is, originally, its DIFFERENTIAL (a FIELD of a one-form). However, there is a standard way to convert it into a VECTOR FIELD (which is also called a gradient of that function).