École Internationale Polytechnique de Genève (EIPG) e-Learning platform (LMS & LCMS)
La plateforme supporte 26 langues, et respecte les normes SCORM 1.2 et 2004, et est compatible avec divers modèle didacticiel, dont: apprentissage mixte, auto-formation, formation collaborative, et même apprentissage social via le Chat, le Wiki, les forums et d'autres fonctions différentes.
BVMDAE (examen de validation de License / accès aux maîtrise et au doctorat)
Vous êtes en possession d'un Maîtrise ou Doctorat d'autres institutions, vous devez toujours passer le BVMDAE.
![BVMDAE examinations](eipg/courses/BVMDAE.jpg)
Que vous ayez étudié à Oxford, Harvard, Cambridge, Yale, Open University, UNISA, SAIM, IBS, ICS, Intec, Alison, Coursera ou Fun MOOC, vous devez passer le BVMDAE (Bachelor Validation / Master & Doctorate's Access Exam). Le BVMDAE deviendra le futur outil d'évaluation du niveau éducatif. Le BVMDAE a pour objectif de maintenir le plus haut niveau possible. Prenez note que les étudiants d'Oxford, Harvard, Cambridge et Yale ont du mal à réussir le BVMDAE.
L'enseignement professionnel en ligne EIPG (et ce qui est ...) peut avoir lieu au niveau postsecondaire, formation continue et d'enseignement supérieur; et peut interagir avec le système d'apprentissage. Au niveau postsecondaire, l'enseignement professionnel est souvent dispensé par des métiers hautement spécialisés, des écoles techniques, des collèges communautaires, des collèges d'enseignement supérieur du Royaume-Uni, des universités, des instituts de technologie / instituts polytechniques, etc.
Jusqu'à récemment, la quasi-totalité de l'enseignement professionnel se déroulait en classe ou sur le lieu de travail, les étudiants apprenant les compétences professionnelles et la théorie du métier auprès de professeurs accrédités ou de professionnels établis. Cependant, la formation professionnelle en ligne a gagné en popularité et a permis plus facilement que jamais aux étudiants d'acquérir diverses compétences professionnelles et compétences générales auprès de professionnels établis de l'industrie.
Qu'en est-il de la reconnaissance ... Le BVMDAE (examen de validation de License / accès aux maîtrise et au doctorat) sert d'examen de reconnaissance automatisé. La plupart des étudiants de l'Université publique échouent à cet examen, ne prenez donc pas cet examen pour acquis.
Faculty of Management
Doctor Of Science Business Management & Administration or D.Sc. BMA
will be of interest to persons wanting to develop skills in Management, Administration, Managerial science and Entrepreneur in general.
![D.Sc. BMA courses](eipg/courses/dbma.jpg)
Faculty of Medecine
Doctor Of Science Natural Medicine or D.Sc. nMed.
studing anatomy, physiology, biomedical science and medicine in general, as well as any persons wanting to expand their knowledge and understanding of Natural Medicine.
![Medicine courses](eipg/courses/research-procedures.jpg)
Doctor Of Science Nursing, Nutrition & Naturopathy or D.Sc. NNN
nursing students will study anatomy, physiology, biomedical science and nursing in general.
![Nursing courses](eipg/courses/free-nurse-continuing-education.jpg)
Doctor Of Science Psychiatrist & Naturopathy or D.Sc. PSY
is a physician who specializes in psychiatry, the branch of medicine devoted to the diagnosis, prevention, study, and treatment of mental disorders.
![Psychiatrist courses](eipg/courses/Psychiatrist.jpg)
Doctor Of Science Psychology & Naturopathy or D.Sc. PGY
... attempt to understand the role of mental functions in individual and social behavior, while also exploring the physiological and biological processes that underlie cognitive functions and behaviors.
![Psychology courses](eipg/courses/psychologist.jpg)
Doctor Of Science Human Health & Nutrition or D.Sc. HHN
... Human nutrition deals with the provision of essential nutrients in food that are necessary to support human life and health. Poor nutrition is a chronic problem often linked to poverty, food security or a poor understanding of nutrition and dietary practices. Malnutrition and its consequences are large contributors to deaths and disabilities worldwide. Good nutrition is necessary for children to grow physically, and for normal human biological development.
![Nuritionist courses](eipg/courses/nutritionist.jpg)
Doctor of Science Biotechnology or D.Sc. BTECH
BioTech or BTCH, also referred to as Biotechnology, is a broad area of biology, involving the use of living systems and organisms to develop or make products. Depending on the tools and applications, it often overlaps with related scientific fields. In the late 20th and early 21st centuries, biotechnology has expanded to include new and diverse sciences, such as genomics, recombinant gene techniques, applied immunology, and development of pharmaceutical therapies and diagnostic tests. The term biotechnology was first used by Karl Ereky in 1919, meaning the production of products from raw materials with the aid of living organisms.
![Biotechnology courses](eipg/courses/biotechnology.gif)
Doctor of Science Biomedical Engineering or D.Sc. BME
Biological medical engineering or bioengineering, BioMed or BME, is the application of principles of biology and the tools of engineering to create usable, tangible, economically-viable products. Biological engineering employs knowledge and expertise from a number of pure and applied sciences, such as mass and heat transfer, kinetics, biocatalysts, biomechanics, bioinformatics, separation and purification processes, bioreactor design, surface science, fluid mechanics, thermodynamics, and polymer science. It is used in the design of medical devices, diagnostic equipment, biocompatible materials, renewable energy, ecological engineering, agricultural engineering, process engineering and catalysis, and other areas that improve the living standards of societies.
![Bio Medical Engineering courses](eipg/courses/stingray-duo.jpg)
Faculty of Science
Doctor of Science Astronomy & Space Science or D.Sc. ASS
is the scientific study of celestial objects (such as stars, planets, comets, and galaxies) and phenomena that originate outside the Earth's atmosphere (such as the cosmic background radiation).
![Our micro-satellite looks like this one (Illustrations) with our Gyroscope, Geiger Counter, Thermostat and other Scientific Instrument for Pressure / Vacuum Measurement](eipg/courses/PRISM_AutoF.jpg)
EIPG Orbiter II - micro-satellite
Our micro-satellite(Illustrations) with Gyroscope, Geiger Counter, Thermostat and other Scientific Instrument for Pressure / Vacuum Measurement, Electromagnetism, Gas Composition etc. Fully Designed and Developed by Prof. Horstmann and his team of researchers. Cost per unit launched into space 12 000.00€
Doctor of Science Chemistry or D.Sc. CHY
is the scientific discipline involved with elements and compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they undergo during a reaction with other substances.
![Chemistry courses](eipg/courses/chemistry.jpg)
Doctor of Science Ecology & Evolution or D.Sc. EYE
Ecology is the scientific analysis and study of interactions among organisms and their environment. Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes that are passed on from parent to offspring during reproduction. Different characteristics tend to exist within any given population as a result of mutation, genetic recombination and other sources of genetic variation.
![Ecology and Evolution courses](eipg/courses/Ecology-Evolution.jpg)
Doctor of Science Environmental Science & Engineering or D.Sc. ESE
Environmental science is an interdisciplinary academic field that integrates physical, biological and information sciences (including ecology, biology, physics, chemistry, plant science, zoology, mineralogy, oceanography, limnology, soil science, geology and physical geography (geodesy), and atmospheric science) to the study of the environment, and the solution of environmental problems.
Environmental engineering is a professional engineering discipline that encompasses broad scientific topics like chemistry, biology, ecology, geology, hydraulics, hydrology, microbiology, and mathematics to create solutions that will protect and also improve the health of living organisms and improve the quality of the environment. Environmental engineering is a sub-discipline of civil engineering and chemical engineering.
![Environmental courses](eipg/courses/EnvironmentalScience.jpg)
Doctor of Science General Science or D.Sc. GSC
... is science in it's basic and pure form. The science which deals with the things related to our life. It includes Biology, chemistry, physics, mathematics and environmental science.
![General Science courses](eipg/courses/maxresdefault.jpg)
Doctor of Science Humanity & Social Science or D.Sc. HSS
... are the study of human behaviour and interaction in social, cultural, environmental, economic and political contexts.
![Humanities and Social Science courses](eipg/courses/Humanities-and-Social-Science.jpg)
Doctor of Science Life Sciences or D.Sc. LSC
the life sciences or biological sciences comprise the branches of science that involve the scientific study of life and organisms – such as microorganisms, plants, and animals including human beings.
![Life Science courses](eipg/courses/LifeScience.jpg)
Doctor of Science Mathematics or D.Sc. MTH
from Greek μάθημα máthēma, "knowledge, study, learning", includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (mathematical analysis). It has no generally accepted definition.
![Mathematics courses](eipg/courses/Northwest-Corner-Method-Exa.jpg)
Doctor of Science Physics or D.Sc. PHY
is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force.
![Physics courses](eipg/courses/model-physics.jpg)
Doctor of Science Statistics or D.Sc. STA
... data collection, organization, analysis, interpretation and presentation. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied.
![Statistics courses](eipg/courses/Statistics.jpg)
Faculty of Engineering
Doctor Of Science Aeronautics & Astronautics Engineering or D.Sc. AAE
is the primary field of engineering concerned with the development of aircraft and spacecraft. It has two major and overlapping branches: aeronautical engineering and astronautical engineering.
Water fueled jet Engine
![Aerospace Engineering courses](eipg/courses/jet-engines.jpg)
Liquid rocket motors
![Aerospace Engineering courses](eipg/courses/rocket.jpg)
Doctor of Science Avionics Engineering or D.Sc. AVE
avionics engineers design and develop aircraft and spacecraft avionic instrumentation. They also conduct research to address problems associated with flight safety systems, landing gear and electronic navigation systems.
![Avionics Engineering courses](eipg/courses/Boeing_787-8_N787BA_cockpit.jpg)
Doctor of Science Agricultural Engineering or D.Sc. AGE
is the area of engineering concerned with the design, construction and improvement of farming equipment and machinery. Agricultural engineers integrate technology with farming.
![Agriculture courses](eipg/courses/agriculture.jpg)
Doctor of Science Civil Engineering or D.Sc. CE
is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built ...
![Civil Engineering courses](eipg/courses/online-civil-engineering-courses.jpg)
Doctor of Science Chemical Engineering or D.Sc. CHE
engineering is a branch of engineering that uses principles of chemistry, physics, mathematics, biology, and economics to efficiently use, produce, design, transport and transform energy and materials.
![Chemical Engineering courses](eipg/courses/chemical-wallpapers-8.jpg)
Doctor of Science Control & Instrumentation Engineering or D.Sc. CIE
Instrumentation is defined as the art and science of measurement and control of the process variables within a production or manufacturing area.
![Instrument Engineering courses](eipg/courses/instrumentation.jpg)
Doctor of Science Design Engineering or D.Sc. DSE
is a person who may be involved in any of various engineering disciplines including civil, mechanical, electrical, chemical, textiles, aerospace, nuclear, manufacturing, systems, and structural building/architectural.
![Design Engineering courses](eipg/courses/design-engineer.jpg)
Doctor of Science Electronics and Communication Engineering or D.Sc. ECE
is the application of science and mathematics to practical problems in the field of electronics and communications. Electronics and communications engineers engage in research, design, development and testing of the electronic equipment used in various systems.
![Electronic Communication Engineer courses](eipg/courses/junction-diagram.jpg)
Doctor of Science Electrical & Computer Engineering or D.Sc. EE
... apply electrical, electronic and magnetic theory to obtain solutions for problems related to the development, design and operation of electrical hardware and software, control systems, electrical machines and communications systems.
![Electrical Engineering courses](eipg/courses/student-pwr.gif)
Doctor of Science Engineering Sciences or D.Sc. ESC
is the application of scientific, economic, social, and practical knowledge in order to design, build, and maintain structures, machines, devices, systems, materials and processes.
![Engineering Sciences courses](eipg/courses/EngineeringSciences.jpg)
Doctor of Science Geology and Geophysics Engineering or D.Sc. GGE
is the study of geologic structures such as faults and folds. Tectonics deals with large scale Earth structures such as tectonic plates and its deep interior. Geophysics is the study of the Earth by quantitative physical methods. It is an applied science and includes the Earth’s interior, crust, oceans, atmosphere and magnetosphere.
![Geology and Geophysics Engineering courses](eipg/courses/GeoDisciplines.jpg)
Doctor of Science Information Systems Engineer or D.Sc. ISE
design, implement, and maintain computer systems that collect, process, store, and present data.
![Information Systems Engineer courses](eipg/courses/all-in-one-pc-21-mos64atom.jpg)
Doctor of Science Mechanical Engineering or D.Sc. ME
is the discipline that applies engineering, physics, engineering mathematics, and materials science principles to design, analyze, manufacture, and maintain mechanical systems.
![Mechanical courses](eipg/courses/8_forward_curved_blades.jpg)
Doctor of Science Mining Engineering or D.Sc. MNE
is an engineering discipline that applies science and technology to the extraction of minerals from the earth.
![Mining Engineering courses](eipg/courses/Mine-Engineer.jpg)
Doctor of Science Materials & Metallurgical Engineering or D.Sc. MME
is the study of metals. Combining theory and practice, degree programs cover the mining, extraction, design and processing of metals, as well as how metals react to environmental changes or stress.
![Materials & Metallurgic courses](eipg/courses/matmet.jpeg)
Doctor Of Science Materials Science & Engineer or D.Sc. MSE
is the study of metals. Combining theory and practice, degree programs cover the mining, extraction, design and processing of metals, as well as how metals react to environmental changes or stress.
![Materials courses](eipg/courses/mes.jpg)
Doctor of Science Metallurgical Engineering or D.Sc. MTE
is the study of metals. Combining theory and practice, degree programs cover the mining, extraction, design and processing of metals, as well as how metals react to environmental changes or stress.
![Metallurgic courses](eipg/courses/metallurgical.jpg)
Doctor of Science Ocean & Atmospheric Engineering or D.Sc. OAE
... It includes the study of physical processes acting across the ocean boundaries at the sea floor, coast, and sea surface, and of the ocean's role in the Earth's climate system.
![Ocean & Atmospheric Engineering courses](eipg/courses/ocean-engineer.jpg)
Doctor of Science Petroleum Engineering or D.Sc. PTE
is a field of engineering concerned with the activities related to the production of hydrocarbons, which can be either crude oil or natural gas.
![Bio-Petrochemical Engineering courses](eipg/courses/algae-biodiesel-6.jpg)
Doctor of Science Production and Industrial Engineering or D.Sc. PIE
is an interdisciplinary engineering discipline that includes manufacturing technology, engineering sciences, management science, and optimization of complex processes, systems, or organizations.
![Production & Industrial Engineering courses](eipg/courses/Production-Industrial.jpg)
Doctor of Science Textile Engineering and Fibre Science or D.Sc. FTE
the application of scientific and engineering principles to the design and control of all aspects of fiber, textile, and processes, products, and machinery. ... A textile engineer therefore works with textile materials: fibers, yarns, fabrics, and finishes.
![Textile Engineering courses](eipg/courses/Textile-Engineering.jpg)
Doctor of Science Vehicle Engineer (VE) D.Sc. VE
is the design, manufacture and operation of motorcycles, automobiles and trucks and their respective engineering subsystems.
![Vehicle Engineer courses](eipg/courses/car.jpg)
Faculty of Arts
Doctor of Science Architecture & Planning or D.Sc. ARP
is a design and planning for a building, and can contain architectural drawings, specifications of the design, calculations, time planning of the building process, and other documentation.
![Architecture courses](eipg/courses/Architecture-Planning.jpg)
Doctor of Science Archaeology & Anthropology or D.Sc. AAT
archaeology focuses on the study of past human cultures through research of historical sites and artifacts, such as tools, skeletal remains and structural ruins in general.
![Archaeology & Anthropology courses](eipg/courses/ArchaeologyAnthropology.jpg)
Faculty of Theology
Doctor of Science Theology or D.Sc. TH
is the study of religion. ... While some students take theology in preparation for a career in the church, you don't have to be religious to have an interest in religion.
![Theology courses](eipg/courses/biblical-theology.jpg)
Private School - IS Of Jerusalem
From Kindergarten to 12 Grade - Homeschool for Isolated individuals
Diploma of Science High School or Dip.Sc. Hs.
High school graduation is a rite of passage in modern society. Whether you want to impress your friends, get the job of your dreams or qualify for an upcoming promotion, we can help. IS Of Jerusalem has partnered with EIPG Online Academy to complete your primary education with a Diploma. Avoid the stigma attached to GEDs with a diploma that says "I finished High School."
IS Of Jerusalem
![High school students raising hands, in classroom with professor](eipg/courses/High-school-students.jpg)
EIPG for "IS Of Jerusalem" - High school students raising hands, in classroom with professor
General Information ...
Magnesium aluminate, Synthetic spinel, accidentally produced in the mid-18th century, has been more recently described in scientific publications in 2000 and 2004. By 2015, transparent spinel was made into sheets and other forms by sintering. Synthetic spinel, which looks like glass but has a significantly higher resistance to pressure, can also have applications for military and commercial purposes.
![Spinel in different Colors](eipg/courses/spinelB.jpg)
At EIPG our Students produce different Turbo Machines Transparent like Glass, Light as Feathers and Hard as Tanks from Spinel(Magnesium aluminate) formula MgAl2O4.
We are at the point of realizing a 3D Printer for Simply Printing Pumps and PD Pumps(Positive Displacement Pumps) in Spinel at 650°C.
Periodic Table of Chemical Elements
Periodic Table of Chemical Elements for EIPG students
![Periodic Table of Chemical Elements](eipg/courses/periodic-table-of-chemical-elements.jpg)
![Girl in a jacket](eipg/courses/cerveau.jpg)
Graphs on all pages are now functional thanks to Prof. Horstmann's Graph API integration
Beautiful Graphs in all browsers
Math Equations on all pages are now functional thanks to Prof. Horstmann's Math API integration
Beautiful Math Equations in all browsers
Integral Formula
\begin{align} f(a) = \frac{1}{2\pi i} \oint_{\gamma}\frac{f(z)}{z-a}dz \end{align}
Dynamic Equations - step by step
Expand the following: \begin{align} (x 1)^2 &= \cssId{Step1}{(x 1)(x 1)} \\[3px] &\cssId{Step2}{{} = x(x 1) 1(x 1)} \\[3px] &\cssId{Step3}{{} = (x^2 x) (x 1)} \\[3px] &\cssId{Step4}{{} = x^2 (x x) 1} \\[3px] &\cssId{Step5}{{} = x^2 2x 1} \end{align}
The Quadratic Formula
When \(a \ne 0\), there are two solutions to \(ax^2 bx c = 0\) and they are
\begin{align} x = {-b \pm \sqrt{b^2-4ac} \over 2a}. \end{align}
Double angle formula for Cosines
\begin{align} \cos(θ φ)=\cos(θ)\cos(φ)−\sin(θ)\sin(φ) \end{align}
Gauss' Divergence Theorem
\begin{align} \int_D ({\nabla\cdot} F)dV=\int_{\partial D} F\cdot ndS \end{align}
Curl of a Vector Field
\begin{align} \vec{\nabla} \times \vec{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \mathbf{i} \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \mathbf{j} \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \mathbf{k} \end{align}
Standard Deviation
\begin{align} \sigma = \sqrt{ \frac{1}{N} \sum_{i=1}^N (x_i -\mu)^2} \end{align}
Definition of Christoffel Symbols
\begin{align} (\nabla_X Y)^k = X^i (\nabla_i Y)^k = X^i \left( \frac{\partial Y^k}{\partial x^i} \Gamma_{im}^k Y^m \right) \end{align}
The Lorenz Equations
\begin{align} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z xy \end{align}
Cauchy's Integral Formula
\begin{align} f(a) = \frac{1}{2\pi i} \oint\frac{f(z)}{z-a}dz \end{align}
The Cauchy-Schwarz Inequality
\[ \left( \sum_{k=1}^n a_k b_k \right)^{\!\!2} \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) \]
A Cross Product Formula
\[ \mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\ \end{vmatrix} \]
The probability of getting \(k\) heads when flipping \(n\) coins is:
\[P(E) = {n \choose k} p^k (1-p)^{ n-k} \]
An Identity of Ramanujan
\[ \frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} = 1 \frac{e^{-2\pi}} {1 \frac{e^{-4\pi}} {1 \frac{e^{-6\pi}} {1 \frac{e^{-8\pi}} {1 \ldots} } } } \]
A Rogers-Ramanujan Identity
\[ 1 \frac{q^2}{(1-q)} \frac{q^6}{(1-q)(1-q^2)} \cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j 2})(1-q^{5j 3})}, \quad\quad \text{for $|q|<1$}. \]
Maxwell's Equations
\begin{align} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, \, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{align}
In-line Mathematics
Finally, while display equations look good for a page of samples, the ability to mix math and text in a paragraph is also important. This expression \(\sqrt{3x-1} (1 x)^2\) is an example of an inline equation. As you see, equations can be used this way as well, without unduly disturbing the spacing between lines.
Trig Identity Formulas
Use these fundemental formulas of trigonometry to help solve problems by re-writing expressions in another equivalent form.
Basic Identities
\[\sin(x)=\frac{1}{\csc(x)}\]
\[\cos(x)=\frac{1}{\sec(x)}\]
\[\tan(x)=\frac{1}{\cot(x)}\]
\[\sec(x)=\frac{1}{\cos(x)}\]
\[\csc(x)=\frac{1}{\sin(x)}\]
\[\cot(x)=\frac{1}{\tan(x)}\]
\[\tan(x)=\frac{\sin(x)}{\cos(x)}\]
\[\sin(-x)=-\sin(x)\]
\[\cos(-x)=\cos(x)\]
\[\tan(-x)=-\tan(x)\]
Pythagorean Identities
\[\sin^2(x) \cos^2(x)=1\]
\[1 \tan^2(x)=\sec^2(x)\]
\[1 \cot^2(x)=\csc^2(x)\]
Sum and Difference Formulas
\[\sin(a b)=\sin(a)\cos(b) \cos(a)\sin(b)\]
\[\sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b)\]
\[\cos(a b)=\cos(a)\cos(b)-\sin(a)\sin(b)\]
\[\cos(a-b)=\cos(a)\cos(b) \sin(a)\sin(b)\]
\[\tan(a b)=\frac{\tan(a) \tan(b)}{1-\tan(a)\tan(b)}\]
\[\tan(a-b)=\frac{\tan(a)-\tan(b)}{1 \tan(a)\tan(b)}\]
\[\sin(x) \sin(y)=2\sin(\frac{x y}{2})\cos(\frac{x-y}{2})\]
\[\sin(x)-\sin(y)=2\cos(\frac{x y}{2})\sin(\frac{x-y}{2})\]
\[\cos(x) \cos(y)=2\cos(\frac{x y}{2})\cos(\frac{x-y}{2})\]
\[\cos(x)-\cos(y)=-2\sin(\frac{x y}{2})\sin(\frac{x-y}{2})\]
Double Angle Formulas
\[\sin(2x)=2\sin(x)\cos(x)\]
\[\cos(2x)=\cos^2(x)-\sin^2(x)=1-2\sin^2(x) = 2\cos^2(x)-1\]
Half Angle Formulas
\[\sin(\frac{x}{2})=\pm\sqrt{\frac{1-\cos(x)}{2}}\]
\[\cos(\frac{x}{2})=\pm\sqrt{\frac{1 \cos(x)}{2}}\]
\[\tan(\frac{x}{2})=\pm\sqrt{\frac{1-\cos(x)}{1 \cos(x)}}=\frac{1-\cos(x)}{\sin(x)}=\frac{\sin(x)}{1 \cos(x)}\]
Trigonometric Products
\[\sin(x)\cos(y)=\frac{\sin(x y) \sin(x-y)}{2}\]
\[\cos(x)\cos(y)=\frac{\cos(x y) \cos(x-y)}{2}\]
\[\sin(x)\sin(y)=\frac{\cos(x-y)-\cos(x y)}{2}\]
Chemical Equations on all pages are now functional thanks to Prof. Horstmann's Chemical API integration
Beautiful Chemical Equations in all browsers
$$\ce{Zn^2 <=>[ 2OH-][ 2H ] $\underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}}$$ <=>[ 2OH-][ 2H ] $\underset{\text{Hydroxozikat}}{\ce{[Zn(OH)4]^2-}}$$}$$ Organic chemistry molecule ...
$$\ce{CH3\bond{1}} {\stackrel{ \;\;\;\large\ce{CH3} }{\stackrel{|}{\underset{\underset{\huge\ce{Cl}}{|}}{\ce{C}}}}}\ce{-CH3}$$
Chemical Equations (ce) ...
$$\ce{CO2 C -> 2 CO}$$
$$\ce{Hg^2 ->[I-] HgI2 ->[I-] [Hg^{II}I4]^2-}$$
$$C_p[\ce{H2O(l)}] = \pu{75.3 J // mol K}$$
Chemical Formulae
$$\ce{H2O}$$ $$\ce{Sb2O3}$$
Charges
$$\ce{H }$$ $$\ce{CrO4^2-}$$ $$\ce{[AgCl2]-}$$
$$\ce{Y^99 }$$ $$\ce{Y^{99 }}$$
Stoichiometric Numbers
$$\ce{2 H2O}$$ $$\ce{2H2O}$$ $$\ce{0.5 H2O}$$
$$\ce{1/2 H2O}$$ $$\ce{(1/2) H2O}$$ $$\ce{$n$ H2O}$$
Isotopes
$$\ce{^{227}_{90}Th }$$ $$\ce{^227_90Th }$$ $$\ce{^{0}_{-1}n^{-}}$$ $$\ce{^0_-1n-}$$ $$\ce{H{}^3HO}$$ $$\ce{H^3HO}$$
Reaction Arrows
$$\ce{A -> B}$$ $$\ce{A <- B}$$ $$\ce{A <-> B}$$ $$\ce{A <--> B}$$ $$\ce{A <=> B}$$ $$\ce{A <=>> B}$$ $$\ce{A <<=> B}$$ $$\ce{A ->[H2O] B}$$ $$\ce{A ->[{text above}][{text below}] B}$$ $$\ce{A ->[$x$][$x_i$] B}$$
Parentheses, Brackets, Braces
$$\ce{(NH4)2S}$$ $$\ce{[\{(X2)3\}2]^3 }$$ $$\ce{CH4 2 $\left( \ce{O2 79/21 N2} \right)$}$$
States of Aggregation
$$\ce{H2(aq)}$$ $$\ce{CO3^2-_{(aq)}}$$ $$\ce{NaOH(aq,$\infty$)}$$
Crystal Systems
$$\ce{ZnS($c$)}$$ $$\ce{ZnS(\ca$c$)}$$
Variables like x, n, 2n 1
$$\ce{NO_x}$$ $$\ce{Fe^n }$$ $$\ce{x Na(NH4)HPO4 ->[\Delta] (NaPO3)_x x NH3 ^ x H2O}$$
Greek Characters
$$\ce{\mu-Cl}$$ $$\ce{[Pt(\eta^2-C2H4)Cl3]-}$$ $$\ce{\beta }$$ $$\ce{^40_18Ar \gamma{} \nu_e}$$
(Italic) Math
$$\ce{NaOH(aq,$\infty$)}$$ $$\ce{Fe(CN)_{$\frac{6}{2}$$}}$$ $$\ce{X_{$i$}^{$x$}}$$ $$\ce{X_$i$^$x$}$$
Italic Text
$$\ce{$cis${-}[PtCl2(NH3)2]}$$ $$\ce{CuS($hP12$)}$$
Upright Text, Escape Parsing
$$\ce{{Gluconic Acid} H2O2}$$ $$\ce{X_{{red}}}$$ $$\ce{{( )}_589{-}[Co(en)3]Cl3}$$
Bonds
$$\ce{C6H5-CHO}$$ $$\ce{A-B=C#D}$$ $$\ce{A\bond{-}B\bond{=}C\bond{#}D}$$ $$\ce{A\bond{1}B\bond{2}C\bond{3}D}$$ $$\ce{A\bond{~}B\bond{~-}C}$$ $$\ce{A\bond{~--}B\bond{~=}C\bond{-~-}D}$$ $$\ce{A\bond{...}B\bond{....}C}$$ $$\ce{A\bond{->}B\bond{<-}C}$$
Addition Compounds
$$\ce{KCr(SO4)2*12H2O}$$ $$\ce{KCr(SO4)2.12H2O}$$ $$\ce{KCr(SO4)2 * 12 H2O}$$
Oxidation States
$$\ce{Fe^{II}Fe^{III}2O4}$$
Unpaired Electrons, Radical Dots
$$\ce{OCO^{.-}}$$ $$\ce{NO^{(2.)-}}$$
Kröger-Vink Notation
$$\ce{Li^x_{Li,1-2x}Mg^._{Li,x}$$V'_{Li,x}Cl^x_{Cl}}$$ $$\ce{O''_{i,x}}$$ $$\ce{M^{..}_i}$$ $$\ce{$V$^{4'}_{Ti}}$$ $$\ce{V_{V,1}C_{C,0.8}$$V_{C,0.2}}$$
Equation Operators
$$\ce{A B}$$ $$\ce{A - B}$$ $$\ce{A = B}$$ $$\ce{A \pm B}$$
Precipitate and Gas
$$\ce{SO4^2- Ba^2 -> BaSO4 v}$$ $$\ce{A v B (v) -> B ^ B (^)}$$
Other Symbols and Shortcuts
$$\ce{NO^*}$$ $$\ce{1s^2-N}$$ $$\ce{n-Pr}$$ $$\ce{iPr}$$ $$\ce{\ca Fe}$$ $$\ce{A, B, C; F}$$ $$\ce{{and others}}$$
Complex Examples
$$\ce{Zn^2 <=>[ 2OH-][ 2H ] $\underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}}$$ <=>[ 2OH-][ 2H ] $\underset{\text{Hydroxozikat}}{\ce{[Zn(OH)4]^2-}}$$}$$ $$K = \frac{[\ce{Hg^2 }][\ce{Hg}]}{[\ce{Hg2^2 }]}$$ $$K = \ce{\frac{[Hg^2 ][Hg]}{[Hg2^2 ]}}$$ $$\ce{Hg^2 ->[I-] $\underset{\mathrm{red}}{\ce{HgI2}}$$ ->[I-] $\underset{\mathrm{red}}{\ce{[Hg^{II}I4]^2-}}$$}$$
Physical Units (pu)
$$\pu{123 kJ}$$ $$\pu{123 mm2}$$ $$\pu{123 J s}$$ $$\pu{123 J*s}$$ $$\pu{123 kJ/mol}$$ $$\pu{123 kJ//mol}$$ $$\pu{123 kJ mol-1}$$ $$\pu{123 kJ*mol-1}$$ $$\pu{1.2e3 kJ}$$ $$\pu{1,2e3 kJ}$$ $$\pu{1.2E3 kJ}$$ $$\pu{1,2E3 kJ}$$
Crystal Systems
$$\ce{ZnS($c$)}$$ $$\ce{ZnS(\ca$c$)}$$
In-line Chemical Equations
Finally, while display equations look good for a page of samples, the ability to mix Chemical Equations and text in a paragraph is also important. This expression \(\ce{Zn^2 <=>[ 2OH-][ 2H ] $\underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}}$$ <=>[ 2OH-][ 2H ] $\underset{\text{Hydroxozikat}}{\ce{[Zn(OH)4]^2-}}$$}\) is an example of an inline equation. As you see, equations can be used this way as well, without unduly disturbing the spacing between lines.