Advanced Mathematics Personal Study Notes: Part I

The thought of publishing my personal study notes that covered fifteen years of undergraduate, graduate, and postdoctoral learning and research dates to the year 1969. The only difference between my forty year old thought and the present is the phenomenal invention and growth of the Internet and computer software. Then, our copy machines were run by ammonia deposits on chemically treated paper. The stencil tissues and engravers were the common tools for small size publication. None of...
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The thought of publishing my personal study notes that covered fifteen years of undergraduate, graduate, and postdoctoral learning and research dates to the year 1969. The only difference between my forty year old thought and the present is the phenomenal invention and growth of the Internet and computer software. Then, our copy machines were run by ammonia deposits on chemically treated paper. The stencil tissues and engravers were the common tools for small size publication. None of those was economically viable during my years as student with total dependency on scholarships and family support. In 1986, after settling in Colorado Springs, Colorado, my elder sister volunteered to ship all my belonging from Egypt to the USA. A half-ton wooden box traveled from Alexandria, Egypt, to Houston, Texas, carrying all my handwritten notes and college textbooks. Then, the custom clearance and transportation between Texas and Colorado ran under $500. The kindness of my sister for risking the enormous troubles in dealing with the Egyptian bureaucratic exporting agencies was paralleled with the similar kindness of the America truck driver who had to drive through a residential apartment complex with an 18-wheeler, yet to find out that his truck was not equipped with a lift that could lower the half-ton container on the ground of the apartment complex. The driver allowed me an extra day to ready and leased fork lift from a local rental shop. Driving a forklift on the streets of Colorado Springs was one of many adventures I enjoyed in such beautiful Rocky town.By 1986, I already gathered more modern and elegant books in mathematics. But, my personal notes embodied a process of engraving the mathematical thinking into my neurons. My cursive writings and graphic illustrations during interactive learning with lecturers and independent home searching and preparation are vividly recorded into forty year old notes. Those include the theoretical subject matter, the examples that followed, and the challenging problems for exercising. I strove to keep the notes as complete as they developed during the learning process. My graphic approach to the interpretation of mathematically abstract concepts illustrates the mental development of my learning. The present study notes comprise linkable topics with bitmapped snips intermingled with searchable text and cover the following topics: 1: Gradient, Divergence and Curl2: Gauss's and Stokes' Theorems3: General Orthogonal Curvilinear Coordinates 4: Tensor Analysis5: The Special Theory of Relativity 6: Matrices7: Special functions 8: Fourier transforms....CHAPTER 1: GRADIENT, DIVERGENCE AND CURL1. Scalar and Vector fields1.1. Definitions of scalar and vector fields2. Continuity3. Partial derivatives4. Increments in point functions5. Directional derivatives6. The gradient of a scalar function6.1. Direction of gradient vector6.2. Practical Application of the gradient6.2.1. Electrostatic field 6.2.2. Equipotential surface or level surface6.2.3. Interpretation of the magnitude and direction of the gradient of a scalar field6.2.4. Example of the gradient derivation7. Line integrals7.1. Application of linear integrals8. Conservative Fields8.1. Application on electrostatic field8.2. Gradient of a single valued differentiable function8.3. Gradient of a constant vector9. The divergence of a vector9.1. Interpretation of divergence of a vector10. The curl of a vector10.1. Interpretation of curl of a vector10.2. Alternative definition of the curl of a vector