History of Calculus PDF

**History of Calculus** \- evolved into a powerful tool used in various fields of science and mathematics. \- development of calculus was not a sudden revelation but a gradual accumulation of ideas and technical methods. **What is calculus?** \- Is the **branch of mathematics** that was develope...

**History of Calculus** \- evolved into a powerful tool used in various fields of science and mathematics. \- development of calculus was not a sudden revelation but a gradual accumulation of ideas and technical methods. **What is calculus?** \- Is the **branch of mathematics** that was developed to **study rates of change.** **-** It was made it so that mathematicians were able to **study continuous quantities in relation** to other reference variables, such as time -The term \"calculus\" initially referred **to general mathematical calculations**, but key developments recognized it as a **distinct branch of mathematics.** **Early precursors: Seeds of calculus** \- Early civilization laid the ground work for the concepts that would rather blossom into calculus. 1. **[EGYPT AND BABYLONIA]** \- **introduced integral calculus ideas**, but without rigorous development. \- **Egyptian Moscow papyrus** provides **formulas for volumes and areas**, but some are approximate and not deductively derived. \- **Babylonians** may have **discovered trapezoidal rule during astronomical observations.** 2. **[ANCIENT GREECE]** **- Eudoxus and Archimedes**, used the **method of exhaustion to calculate areas and volumes**, foreshadowing the concept of the limit. They also made **significant use of infinitesimals.** \- **Democritus** being the **first to consider the division of objects into infinite cross-sections.** \- **Archimedes** developed this method and **invented heuristics similar to modern concepts.** However, infinitesimals were not rigorously accepted until the 17th century, when it was **formalized as the method of Indivisibles.** \- **Archimedes** also **found the tangent to a curve** other than a circle **using differential calculus**. 3. **[CHINA]** \- **Liu Hui invented the method of exhaustion in China** in the 4th century AD to **find the area of a circle.** \- **Zu Chongzh**i established **Cavalieri\'s principle** to **find the volume of a sphere** in the 5th century. 4. **[MIDDLE EAST]** -**Hasan Ibn al-Haytham** derived a f**ormula for the sum of fourth powers**, which was used to **calculate the volume of a paraboloid.** 5. **[INDIA]** \- **Bhaskara I**I demonstrated **familiarity with differential calculus** and **introduced Rolle\'s theorem**. \- The **Kerala School of Astronomy and Mathematics in India contributed further to calculus.** **-** **Madhava of Sangamagrama** and his successors developed calculus components like Taylor series and infinite series approximations, but did not fully synthesize them into a unified framework. **Modern Calculus: A Revolution in Mathematics** \- The 17th century marked a **turning point in the history of calculus**, with the independent development of the discipline by **Isaac Newton** and **Gottfried Wilhelm Leibniz.** **Pioneers of Calculus** - **[ISAAC NEWTON]** \- **English physicist and mathematician,** developed **calculus during the 1660s**, though he did not publish his work until later. -He **used calculus to solve problems in physics**, particularly in his groundbreaking work **Philosophiæ Naturalis Principia Mathematica (1687).** \- He viewed calculus as the **scientific description of the generation of motion and magnitudes.** \- Newton\'s approach **focused on the concept of fluxions,** which **represented instantaneous rates of change.** - **[GOTTFRIED WILHELM LEIBNIZ]** \- **German mathematician and philosopher**, also developed calculus independently around the same time. \- He **published his findings in the 1680s,** introducing a **more systematic notation and rules for working with infinitesimals.** \- His approach **emphasized the concept of differentials, representing infinitely small increments.** \- Leibniz **focused on the tangent problem **and came to believe that **calculus was a metaphysical explanation of change.** - Newton and Leibniz\'s calculus revolutionized mathematics, aiding in physics, engineering, and other fields, but its methods were not always rigorous, sparking debates about its foundations - In the 18th and 19th centuries, mathematicians like **Augustin-Louis Cauchy, Karl Weierstrass, and Bernhard Riemann** reformulated calculus in terms of limits, eliminating the need for infinitesimals. \- This approach provided a more rigorous and consistent framework for calculus, paving the way for its further development and establishing a solid mathematical foundation for modern calculus.