Calculus

Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science,economics, and engineering and can solve many problems for which algebra alone is insufficient.

Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus.

Topics in calculus

Fundamental theorem Limits of functions Continuity Vector calculus Matrix calculus Mean value theorem

Differentiation

Product rule Quotient rule Chain rule Change of variables Implicit differentiation Taylor's theorem Related rates List of differentiation identities

Integration

Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order

Mainstream theories

Mainstream theories (sometimes referred to as central theories) are the body of knowledge of both factual and scientific views and possess a usual scientific quality of the tests of repeatability, consistency with existing well-established science and experimentation. There do exist mainstream theories that are generally accepted theories based solely upon their effects explaining a wide variety of data, although the detection, explanation and possible composition are subjects of debate.

Examples

• Black hole thermodynamics
• Classical mechanics
• Condensed matter physics
• Conservation of energy
• Dark Energy
• Dark matter
• Dynamics
• Electromagnetism
• Field theory
• Fluid dynamics
• General relativity
• Molecular modeling
• Particle physics
• Physical cosmology
• Quantum chromodynamics
• Quantum computers
• Quantum electrochemistry
• Quantum electrodynamics
• Quantum field theory
• Quantum information theory
• Quantum mechanics
• Solid mechanics
• Solid state physics or Condensed Matter Physicsand the electronic structure of materials
• Special relativity
• Standard Model
• Statistical mechanics
• Thermodynamics

PENGANTAR FISIKA MATEMATIKA

PENGANTAR PENGANTAR FISIKA MATEMATIK

Oleh: Dr. Eng. RINTO ANUGRAHA

JURUSAN FISIKA FMIPA UGM  

YOGYAKARTA 

DAFTAR ISI  iii 

BAB I BILANGAN KOMPLEKS   1 

  Beberapa Sifat Aljabar Bilangan Kompleks  4 

  Perkalian dan Pemangkatan, Rumus de Moivre dan Euler 10 

  Rumus Binomium Newton  14 

  Penerapan Bilangan Kompleks  22 

   Mekanika  22 

   Osilator Selaras Teredam  23 

   Masalah Kelistrikan  26 

   Optika  28 

BAB II ALJABAR VEKTOR   32 

  Sifat-Sifat Skalar dan Vektor  32 

  Besar Vektor  33 

  Sifat-Sifat Ruang Vektor  33 

  Penjumlahan Vektor  34 

  Perkalian Antara Vektor  36 

  Perkalian Skalar  36 

  Perkalian Vektor/Silang  40 

  Delta dan Epsilon Kronecker  42 

  Garis dan Bidang  48 

  Bebas dan Gayut Linear  54 

BAB III MATRIKS, DETERMINAN DAN PERSAMAAN LINEAR  59 

  Operasi Matriks  60  

  Rotasi Sumbu-sumbu Koordinat  63 

  Determinan  65 

  Rumus Cramer  70 

BAB IV LIMIT, FUNGSI DAN TURUNAN   81 

  Fungsi  81   

  Macam−macam Fungsi Kontinu  84  

  Limit Fungsi  92  

  Sifat−sifat Limit Fungsi  92 

  Turunan Fungsi  94  

  Deret Taylor dan Deret MacLaurin  98  

  Penerapan Turunan  101 

BAB V INTEGRAL  106 

  Integral sebagai Inversi Penurunan (Anti Derivatif)  106 

  Rumus-Rumus Integral Dasar dan Metode Pengintegralan 106 iv

  Pengintegralan Parsial  108 

  Substitusi Variabel  108 

  Metode Pecahan Parsial  109 

  Integral Tertentu (Integral Riemann  113 

  Penerapan Integral Tertentu   116 

   Mencari Luas di bawah Benda Putar  116 

   Volume Benda Putar  117 

   Menentukan Panjang Busur Kurva  118 

  Fungsi Gamma  120 

  Fungsi Beta  125 

BAB VI FUNGSI VARIABEL BANYAK : TURUNAN PARSIAL  132 

  Turunan Parsial  132 

  Diferensial Total  134 

  Dalil Rantai  138 

  Diferensial Implisit  139  

  Pengubahan Variabel  144 

  Transformasi Legendre    147 

  Ekstremum Fungsi Dua Variabel  150 

DAFTAR PUSTAKA  155

Materi Kuliah Pengantar Matematika

Pengantar Matematika termasuk Mata Kuliah Dasar di Departemen Matematika UI. Pengantar Matematika memperkenalkan kepada mahasiswa bagaimana membuktikan suatu proporsi dengan memakai logika matematika. Bahasa matematika dan logika matematika sangatlah berbeda dengan pengertian logika pada umumnya. Dalam logika matematika banyak digunakan lambang-lambang tertentu. Sebagai referensi, Anda bisa mengunduh file-file berikut.

1. b_induk.pdf
2. Bab2_fol.pdf
3. Bab3_teori_bil_1.pdf
4. Bab4_teori_bil_2.pdf
5. Func_Series.pdf
6. Inference_Rule.pdf
7. Logika_Proporsi.pdf
8. LogPuzzles.pdf
9. Pembuktian.pdf
10. Pohon_Semantik.pdf
11. Set.pdf
12. Set_12.rtf

File-file tersebut merupakan salah satu referensi bagi mahasiswa yang ingin memahami logika matematika.

Sumber:

http://puputo.blogspot.com/2009/06/file-download-untuk-materi-kuliah.html

Mathematical Methods in the Physical Sciences

Mathematical Methods in the Physical Sciences is a 1966 textbook by mathematician Mary L. Boas intended to develop skills in mathematical problem solving needed for junior to senior-graduate courses in engineering, physics, and chemistry. The book provides a comprehensive survey of analytic techniques and provides careful statements of important theorems while omitting most detailed proofs. Each section contains a large number of problems, with selected answers. Numerical computational approaches using computers are outside the scope of the book.

The book, now in its third edition, is still widely used in university classrooms^[1] and is frequently cited in other textbooks and scientific papers.

Chapters

1. Infinite series, power series
2. Complex numbers
3. Linear algebra
4. Partial differentiation
5. Multiple integrals
6. Vector analysis
7. Fourier series and transforms
8. Ordinary differential equations
9. Calculus of variations
10. Tensor analysis
11. Special functions
12. Series solution of differential equations; Legendre, Bessel, Hermite, and Laguerre functions
13. Partial differential equations
14. Functions of a complex variable
15. Integral transforms
16. Probability and statistics

Jika anda mahasiswa, dosen, guru fisika, atau siapapun yang ingin memahami lebih mendalam konsep fisika-matematik, e-book ini sangat tepat menjadi referens. Jika tertarik, anda bisa men-download-nya dengan klik disini.

Sources:

Wikipedia


Ucapan Terima Kasih:

Ibunda Dra. Roswati Mudjiarto, M.Pd.
Pendidikan Fisika Universitas Pendidikan Indonesia

Mathematically Rigorous Physics

The term 'mathematical' physics is also sometimes used in a special sense, to denote research aimed at studying and solving problems inspired by physics within a mathematically rigorous framework. Mathematical physics in this sense covers a very broad area of topics with the common feature that they blend pure mathematics and physics. Although related to theoretical physics, 'mathematical' physics in this sense emphasizes the mathematical rigour of the same type as found in mathematics. 

On the other hand, theoretical physics emphasizes the links to observations and experimental physics which often requires theoretical physicists (and mathematical physicists in the more general sense) to use heuristic, intuitive, and approximate arguments. Such arguments are not considered rigorous by mathematicians. Arguably, rigorous mathematical physics is closer to mathematics, and theoretical physics is closer to physics. This also has an institutional side: Many mathematical physicists are members of mathematics departments.

Such mathematical physicists primarily expand and elucidate physical theories. Because of the required rigor, these researchers often deal with questions that theoretical physicists have considered to already be solved. However, they can sometimes show (but neither commonly nor easily) that the previous solution was incorrect.

The field has concentrated in four main areas:

1. quantum field theory, especially the precise construction of models;
2. statistical mechanics, especially the theory of phase transitions; and
3. nonrelativistic quantum mechanics (Schrödinger operators), including the connections to atomic and molecular physics.
4. quantum information theory

The effort to put physical theories on a mathematically rigorous footing has inspired many mathematical developments. For example, the development of quantum mechanics and some aspects of functional analysis parallel each other in many ways. The mathematical study of quantum statistical mechanics has motivated results in operator algebras. The attempt to construct a rigorous quantum field theory has brought about progress in fields such as representation theory. Use of geometry and topology plays an important role in string theory.

Mathematical Physics

Mathematical physics refers to development of mathematical methods for application to problems in physics.

Scope of the subject

The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories.". There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. The theory of partial differential equations (and the related areas of variational calculus, Fourier analysis, potential theory, and vector analysis) are perhaps most closely associated with mathematical physics.

These were developed intensively from the second half of the eighteenth century (by, for example, D'Alembert, Euler, and Lagrange) until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, elasticity theory, acoustics,thermodynamics, electricity, magnetism, and aerodynamics.

The theory of atomic spectra (and, later, quantum mechanics) developed almost concurrently with the mathematical fields of linear algebra, thespectral theory of operators, and more broadly, functional analysis. These constitute the mathematical basis of another branch of mathematical physics.

The special and general theories of relativity require a rather different type of mathematics. This was group theory: and it played an important role in both quantum field theory and differential geometry. This was, however, gradually supplemented by topology in the mathematical description of cosmological as well as quantum field theory phenomena.

Statistical mechanics forms a separate field, which is closely related with the more mathematical ergodic theory and some parts of probability theory.

There are increasing interactions between combinatorics and physics, in particular statistical physics.

The usage of the term 'Mathematical physics' is sometimes idiosyncratic. Certain parts of mathematics that initially arose from the development of physics are not considered parts of mathematical physics, while other closely related fields are. For example, ordinary differential equationsand symplectic geometry are generally viewed as purely mathematical disciplines, whereas dynamical systems and Hamiltonian mechanics belong to mathematical physics.

Mathematical Methods for Physicists: A concise introduction

Author: TAI L. CHOW

was born and raised in China. He received a BS degree in physics from the National Taiwan University, a Masters degree in physics from Case Western Reserve University, and a PhD in physics from the University of Rochester. Since 1969, Dr Chow has been in the Department of Physics at California State University, Stanislaus, and served as department chairman for 17 years, until 1992. He served as Visiting Professor of Physics at University of California (at Davis and Berkeley) during his sabbatical years. He also worked as Summer Faculty Research Fellow at Stanford University and at NASA. Dr Chow has published more than 35 articles in physics journals and is the author of two textbooks and a solutions manual.

Sumber:

1. Wikipedia
2. Google book

Ucapan Terima Kasih

Kepada Ibunda Roswati Mudjiarto