Calculus II

Calculus is a branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences. The two main types are differential calculus and integral calculus.

In multivariable calculus, we deal with the following topics:

  • Three- and n-dimensional coordinate systems

  • Vectors and vector spaces

  • Dot and cross products

  • Lines and planes in three-dimensional space

  • Quadratic surfaces

  • Curves in space and their arc length

  • Tangent, normal, and binormal unit vectors

  • Curvature and torsion of curves in space

  • Functions of several variables and their partial derivatives

  • Limits and continuity in higher dimensions

  • The chain rules in multi-variable functions

  • Directional and total derivatives, gradient vectors and their applications

  • Curl and divergence vectors

  • Tangent planes, extreme values, and saddle points

  • Polar, cylindrical, and spherical coordinate systems

  • Double and triple integrals over different regions

  • Line integrals and Green's theorem

  • Surface integrals and Stocks' theorem

  • The divergence theorem

For more details, refer to the following book:

Thomas, G. B., Weir, M. D., Hass, J., & Giordano, F. R. (2005). Thomas' calculus. Addison-Wesley.

I also recommend the following book:

Differential and Integral Calculus by Grigorii Mikhailovich Fichtenholz

As one of the founders of the Leningrad (today known as Saint Petersburg) school of "real analysis", Grigorii Mikhailovich Fichtenholz (1888-1959) was a Soviet mathematician. In addition to several books on mathematical analysis, Fichtenholz (also known as Fikhtengolts) wrote a three-volume textbook on differential and integral calculus. This textbook discusses the mathematical analysis of functions of one real variable, functions of many real variables, and complex functions. This well-presented book on mathematical analysis has not yet been translated into English, although it has been translated into Chinese, German, Persian, Polish, and Vietnamese.

The story behind its Persian translation is quite interesting. The late "Bagher Emami باقر امامی" who was a university lecturer in mathematics started a translation of this volumous book. However, the translation was later completed by another person. "While translating almost two volumes of the book, Emami gave his whole work to me and asked me to translate the rest and publish this complete course on mathematical analysis", Ostad "Parviz Shahriari پرویز شهریاری" explains in the preface of the book. Ostad Parviz Shahriari finished the work, but it took some time to find a publisher because producing such a book was expensive. Finally, Ferdos Publications published the book and this significant book on mathematical analysis (differential and integral calculus) became available for Persian speaking people.

Single-variable Calculus

Ordinary Differential Equations

Engineering Mathematics