As in the third part, in the fourth part, I cover the content of the third semester of calculus. To understand the article on the Jacobian, you need to understand what a determinant is, as listed in the section “Further Linear Algebra.” For our purposes, the Jacobian is important in understanding general relativity. Other articles are necessary to understand differential forms and Maxwell’s equations. Curl and Green’s theorem assumes familiarity with line integrals as explained in “Kinetic energy and Potential energy in three dimensions, Line integrals and Gradient” listed in the section “Mathematical Introduction to Physics.” Even though line integrals and gradient can be defined without the aid of physics, I decided to introduce them using physics examples. This is the reason why the article introducing them is listed in the physics section. “Spherical coordinate system” is essential in understanding our later article “Hydrogen atom.”