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# Calculus

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Calculus is the mathematical study of constant change. This is similar to the way that geometry is the study of shapes and algebra is the study of mathematical operations.

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## Ancient & medieval

During both ancient and medieval times, key calculus concepts were established that led to the creation of integral calculus; however, not all of these ideas were fully developed until more recently.

The Egyptian Moscow papyrus, which dates back to the 13th century and is one of the first examples of Egyptian mathematics, includes calculations of both area and volume. However, these calculations are very simple and do not show method; therefore, many of them lack critical components required to fully understand calculus.

Greek mathematician Eudoxus demonstrated the use of the “method of exhaustion,” a precursor to what is known today as a “limit”; something used to calculate area and volume. Later, Greek mathematician and physicist Archimedes further expanded upon this concept, creating heuristics, a technique that closely resembles integral calculus.

Independent of the Greek’s advances in early calculus, the method of exhaustion was also being worked on in 3rd-century China by the scholar Liu Hui who used it to find the area of a circle. In the 5th century, Chinese scholar Zu Gengzhi created a method that is today known as Cavalieri’s principle, which can be used to determine a sphere’s volume.

Sometime between the 10th and 11th centuries, Hasan Ibn al-Haytham, a mathematician and scholar in the Middle East, created a formula for the sum of fourth powers, allowing him to accurately determine the volume of a paraboloid.

Furthermore, in the 14th century, mathematicians in India established a method - similar to that of differentiation - that could be applied to some trigonometric functions. While this method contained components of modern-day calculus, a connection was not drawn between the derivative and integral themes and, therefore, the method could not be further advanced.

## Modern

Modern calculus began to be developed by both Isaac Newton and Gottfried Leibniz (though they were not working together) during the 17th century. However, elements of what is considered modern calculus can also be traced back to societies in Ancient Greece, China, the Middle East, Medieval Europe, and India.

The foundation for which much of modern-day’s calculus is based upon came from the Italian mathematician Bonaventura Cavalieri. He surmised that both volume and area should be calculated as the sums of the areas and volumes of thin cross-section. Despite what we know today, Cavalieri’s results were often riddled with error, resulting in a poor reputation.

Isaac Newton and Gottfried Wilhelm Leibniz, both working completely independently of one another, are most often credited as the developers and creators of modern calculus. Newton, though primarily thought of as a physicist, used various calculus methods to answer the questions of planetary motion and the oblate shape of the earth among others. He was also responsible for creating the fundamental theorem which later branched into the second fundamental theorem (a.k.a. 2nd fundamental theorem).

While Newton was the first person to utilize calculus as a way to solve questions of physics, Leibniz is often credited with developing the majority of calculus notation, which is still used to this day.

## Significance in history

The majority of what is known today in calculus was derived from established ideas that emerged from early civilizations in China, Greece, India, Japan, Iraq, and Persia.

What is known today as calculus today first emerged in 17th-century Europe when both Gottfried Wilhelm Leibniz and Isaac Newton managed to consolidate and develop a number of key basic principles. While Leibniz and Newton are often credited with the creation of modern calculus, they were only able to do this by expanding upon earlier concepts of the area beneath curves and instantaneous motion.

Today, elements of calculus can be applied to discover various values in velocity, acceleration, slope (of a curve), area, volume, center of mass, arc length, and even pressure. The two categories that these calculations fall into are called “differential” and “integral” calculus.

## Applications

All branches of physical science use calculus in some form to solve mathematical problems. This includes computer science, engineering, statistics, medicine, and actuarial science among others.

Calculus is used strongly in physics as all concepts in both electromagnetism and classical mechanics are derived from it. Calculus can also be used to determine an object’s mass, density, total energy, and inertia. For example, Newton’s second law of motion (which states that an object’s momentum is equal to the force acted upon it and will be in the same direction) uses differential calculus to calculate an object’s path.

Einstein’s theory of relativity along with Maxwell’s theory of electromagnetism also both use differential calculus. Chemistry, too, uses calculus to determine things like radioactive decay and reaction rates.

In addition, calculus can be paired with other disciplines (like geometry or linear algebra) in order to calculate solutions.

There are many applications for calculus throughout the field of medicine. Calculus can be used to determine the choicest angle of a blood vessel in order to increase blood flow. It can also be used to determine dosing regulations and laws, and determine the best ways for targeting tumors using radiation therapy.

Today, tools like a calculus cheat sheet or calculus calculator (not to be confused with a calculus bridge - a term used in modern dentistry) can be found on sites like Slader. Additionally, users can find out about stochastic equations, calculus early transcendentals, and get there most pressing questions answered such as “is calculus hard?” and “who is James Stewart, mathematician?”

## Varieties

Various formulations of calculus have been created over time in order to solve specific calculations. Three common formations are known as non-standard calculus, smooth infinitesimal analysis, and constructive analysis (a.k.a. “Constructive mathematics”).

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