Essential Mathematica: With Applications to Mathematics and Physics, based on lecture notes of course taught at the University of Illinois at Chicago for advanced undergrad and graduate students, teaches how to use Mathematica to solve various problems in mathematics and physics. The goal is to provide the reader with Mathematica capabilities quickly and efficiently. The second section includes a variety of applications in physics, engineering and applied mathematics, including Egyptian Fractions, Happy Numbers, Mersenne Numbers, Multibases, Quantum Harmonic Oscillator, Quantum Square Potential, Van der Pol Oscillator, electrostatics, Motion of Charged particle in an electromagnetic field, Duffing Oscillator, Negative and Complex Bases, Curves Tautochrone, Kepler s Law,? Foucault? S Pendulum, iteration Function Systems, Public-Key Encryption, and Julia and Mandelbrot Set. The first part - for example, not long explanations.