A report on Geometry, Calculus, Mathematics, Area and Differential geometry

An illustration of Desargues' theorem, a result in Euclidean and projective geometry
Archimedes used the method of exhaustion to calculate the area under a parabola.
3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
The combined area of these three shapes is approximately 15.57 squares.
A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.
A European and an Arab practicing geometry in the 15th century
Alhazen, 11th-century Arab mathematician and physicist
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
This square and this disk both have the same area (see: squaring the circle).
An osculating circle
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
Isaac Newton developed the use of calculus in his laws of motion and gravitation.
The quadratic formula expresses concisely the solutions of all quadratic equations
A square metre quadrat made of PVC pipe.
An illustration of Euclid's parallel postulate
Gottfried Wilhelm Leibniz was the first to state clearly the rules of calculus.
Rubik's cube: the study of its possible moves is a concrete application of group theory
Although there are 10 mm in 1 cm, there are 100 mm2 in 1 cm2.
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
Maria Gaetana Agnesi
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
The area of this rectangle is lw.
A sphere is a surface that can be defined parametrically (by or implicitly (by x2 + y2 + z2 − r2 = 0.)
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle.
Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metric.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
A parallelogram split into two equal triangles.
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
A page from al-Khwārizmī's Algebra
A circle can be divided into sectors which rearrange to form an approximate parallelogram.
A tiling of the hyperbolic plane
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Archimedes showed that the surface area of a sphere is exactly four times the area of a flat disk of the same radius, and the volume enclosed by the sphere is exactly 2/3 of the volume of a cylinder of the same height and radius.
Differential geometry uses tools from calculus to study problems involving curvature.
Leonhard Euler created and popularized much of the mathematical notation used today.
Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).
A thickening of the trefoil knot
Carl Friedrich Gauss, known as the prince of mathematicians
The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions
Quintic Calabi–Yau threefold
The front side of the Fields Medal
Discrete geometry includes the study of various sphere packings.
The Cayley graph of the free group on two generators a and b
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

Geometry is, with arithmetic, one of the oldest branches of mathematics.

- Geometry

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

- Calculus

Mathematics is an area of knowledge that includes such topics as numbers (arithmetic, number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

- Mathematics

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

- Differential geometry

In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

- Area

Since then, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, and others.

- Geometry

For shapes with curved boundary, calculus is usually required to compute the area.

- Area

Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c.

- Calculus

In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.

- Area

The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions from calculus began around the 1600s when calculus was first developed by Gottfried Leibniz and Isaac Newton.

- Differential geometry

287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of pi.

- Geometry

This was a necessary precursor to the development of calculus and a precise quantitative science of physics.

- Geometry

Such curves can be defined as graph of functions (whose study led to differential geometry).

- Mathematics

Various concepts based on length, such as the arc length of curves, area of plane regions, and volume of solids all possess natural analogues in Riemannian geometry.

- Differential geometry

He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.

- Mathematics

Differential geometry

- Calculus
An illustration of Desargues' theorem, a result in Euclidean and projective geometry

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