Leonhard Euler: Difference between revisions
Jump to navigation
Jump to search
No edit summary |
|||
| (23 intermediate revisions by the same user not shown) | |||
| Line 2: | Line 2: | ||
[[File:Leonhard Euler.jpg|200px|thumb|right|Leonhard Euler]] | [[File:Leonhard Euler.jpg|200px|thumb|right|Leonhard Euler]] | ||
Leonhard Euler was a Swiss Mathematician and physicist who made important contributions to math and physics. Euler is often considered one of the greatest mathematicians to have ever lived. | Leonhard Euler was a Swiss Mathematician and physicist who made important contributions to math and physics. Euler is often considered one of the greatest mathematicians to have ever lived. | ||
| Line 9: | Line 7: | ||
==Early Life== | ==Early Life== | ||
Euler was born in Basel, Switzerland on April 15th, 1707. His father was a minister and the family naturally expected Euler to also go in to ministry. However, his father sparked a curiosity in math for Euler and | Euler was born in Basel, Switzerland on April 15th, 1707. His father was a minister and the family naturally expected Euler to also go in to ministry. However, his father sparked a curiosity in math for Euler and he entered University of Basel at the age of 14, with Johann Bernoulli as his mentor. | ||
==Contributions to Physics== | ==Contributions to Physics== | ||
Arguably the greatest mathematician in history, Euler made lots of contribution to math such as the number | Arguably the greatest mathematician in history, Euler made lots of contribution to math such as the number “<math>e</math>”, the concept of a function, summation notation “Σ”, imaginary unit notation “<math>i</math>”, and popularizing <math title="pi">\pi</math>; however, Euler also made lots of important contributions to physics. | ||
===Euler-Bernoulli beam equation=== | ===Euler-Bernoulli beam equation=== | ||
| Line 19: | Line 17: | ||
:<math>\frac{\mathrm{d}^2}{\mathrm{d} x^2}\left(EI \frac{\mathrm{d}^2 w}{\mathrm{d} x^2}\right) = q\,</math> | :<math>\frac{\mathrm{d}^2}{\mathrm{d} x^2}\left(EI \frac{\mathrm{d}^2 w}{\mathrm{d} x^2}\right) = q\,</math> | ||
where | |||
:<math>w</math> = Out-of-plane displacement of the beam | |||
:<math>E</math> = modulus of elasticity (Young's Modulus), | |||
:<math>I</math> = area moment of inertia, and | |||
:<math>q</math> = distributed load (force per unit length) | |||
===Work in Astronomy=== | ===Work in Astronomy=== | ||
====Understanding the nature of comets==== | ====Understanding the nature of comets==== | ||
In addition to his work with classical mechanics, Euler was recognized by Paris Academy Prizes over the course of his career for calculating, with great accuracy, the orbits of comets and other celestial bodies. | In addition to his work with classical mechanics, Euler was recognized by Paris Academy Prizes over the course of his career for calculating, with great accuracy, the orbits of comets and other celestial bodies. | ||
| Line 27: | Line 30: | ||
====Calculating the parallax of the sun==== | ====Calculating the parallax of the sun==== | ||
[[File:Parallax Example.svg|thumb|300px|right|A simplified illustration of the parallax of an object | [[File:Parallax Example.svg|thumb|300px|right|A simplified illustration of the parallax of an object]] | ||
Euler calculated the parallax of the sun, calculating the difference in the apparent position of the object and the actual position of the object. Euler’s calculation of the parallax later led to the development of more accurate longitude tables. | Euler calculated the parallax of the sun, calculating the difference in the apparent position of the object and the actual position of the object. Euler’s calculation of the parallax later led to the development of more accurate longitude tables. | ||
===Work in optics=== | ===Work in optics=== | ||
[[File:Laser Interference.JPG|thumb|Diffraction pattern of red laser beam | [[File:Laser Interference.JPG|thumb|Diffraction pattern of red laser beam]] | ||
While Newton argued that light was made of particles, Euler argued that light behaved more like waves. In ''Nova theoria lucis et colorum'' (1746), Euler argued that diffractions can be more easily argued with the wave theory rather than the previous “pulse theory”. Euler’s wave theory remained the dominant theory about light until the quantum theory of light. | While Newton argued that light was made of particles, Euler argued that light behaved more like waves. In ''Nova theoria lucis et colorum'' (1746), Euler argued that diffractions can be more easily argued with the wave theory rather than the previous “pulse theory”. Euler’s wave theory remained the dominant theory about light until the quantum theory of light. | ||
===Structural Engineering=== | ===Structural Engineering=== | ||
Euler also published a formula for calculating the force where the strut would fail that is often used in structural engineering. | Euler also published a formula for calculating the force where the strut would fail that is often used in structural engineering, commonly referred to as Euler's Column Formula. | ||
:<math>F=\frac{\pi^2 EI}{(KL)^2}</math> | :<math>F=\frac{\pi^2 EI}{(KL)^2}</math> | ||
| Line 45: | Line 48: | ||
:<math>F</math> = maximum or critical force (vertical load on column), | :<math>F</math> = maximum or critical force (vertical load on column), | ||
:<math>E</math> = modulus of elasticity (Young's Modulus), | :<math>E</math> = modulus of elasticity (Young's Modulus), | ||
:<math>I</math> = area moment of inertia, | :<math>I</math> = area moment of inertia, and | ||
:<math>K L</math> is the length of the column. | |||
:<math>K L</math> is the | |||
===Fluid Dynamics=== | ===Fluid Dynamics=== | ||
Euler in 1757 published a set of equations for flow of an ideal fluid with no viscosity (Inviscid flow) that are now known as the Euler Equations. | Euler in 1757 published a set of equations for flow of an ideal fluid with no viscosity (Inviscid flow) that are now known as the Euler Equations. | ||
:<math> | :<math> | ||
| Line 63: | Line 61: | ||
</math> | </math> | ||
where | |||
:<math>{ \mathbf {u}}</math> = fluid velocity | |||
:<math>p</math> = pressure, and | |||
:<math>\rho</math> = fluid density | |||
== See also == | == See also == | ||
* [[Daniel Bernoulli]] | |||
* [[Wave-Particle Duality]] | |||
* [[Young's Modulus]] | |||
* [[Multisource Interference: Diffraction]] | |||
===Further reading=== | ===Further reading=== | ||
* Dunham, William. <i>Euler: The Master of Us All</i>. Washington, D.C.: Mathematical Association of America, 1999. Print. | |||
===External links=== | ===External links=== | ||
* http://eulerarchive.maa.org/ A digital library dedicated to the work and life of Leonhard Euler | |||
* http://micro.magnet.fsu.edu/optics/timeline/people/euler.html Euler's work on optics | |||
* https://muse.jhu.edu/journals/perspectives_on_science/v016/16.4.pedersen.html Euler's Wave Theory of Light | |||
* http://web.mit.edu/16.unified/www/SPRING/materials/Lectures/M4.7-Unified09.pdf Euler's Column Formula | |||
* http://eulerarchive.maa.org/pages/E088.html ''Nova theoria lucis et colorum'' | |||
==References== | ==References== | ||
* Weisstein, Eric W. "Euler's Equations of Inviscid Motion." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/EulersEquationsofInviscidMotion.html | |||
* "The Euler-Bernoulli Beam." (2005): 368-401. Center for Aerospace Structures. University of Colorado Boulder, 2005. Web. 1 Dec. 2015. | |||
* Grieve, David J. "Euler Buckling Formula Derivation." Euler Buckling Formula Derivation. Plymouth University, 1 Mar. 2004. Web. 01 Dec. 2015. | |||
* Musielak, Dora. "Euler: Genius Blind Astronomer Mathematician." (n.d.): n. pag. ArXiv.org. Cornell University Library, 28 June 2014. Web. 1 Dec. 2015. | |||
[[Category: | [[Category:Notable Scientists]] | ||
Latest revision as of 20:11, 1 December 2015
by Jong Rak Koh
Leonhard Euler was a Swiss Mathematician and physicist who made important contributions to math and physics. Euler is often considered one of the greatest mathematicians to have ever lived.
Early Life
Euler was born in Basel, Switzerland on April 15th, 1707. His father was a minister and the family naturally expected Euler to also go in to ministry. However, his father sparked a curiosity in math for Euler and he entered University of Basel at the age of 14, with Johann Bernoulli as his mentor.
Contributions to Physics
Arguably the greatest mathematician in history, Euler made lots of contribution to math such as the number “[math]\displaystyle{ e }[/math]”, the concept of a function, summation notation “Σ”, imaginary unit notation “[math]\displaystyle{ i }[/math]”, and popularizing [math]\displaystyle{ \pi }[/math]; however, Euler also made lots of important contributions to physics.
Euler-Bernoulli beam equation
The theory validates the beam deflection calculation for laterally loaded beams. The equation provides a relationship between the deflection of the beam and the applied load intensity.
- [math]\displaystyle{ \frac{\mathrm{d}^2}{\mathrm{d} x^2}\left(EI \frac{\mathrm{d}^2 w}{\mathrm{d} x^2}\right) = q\, }[/math]
where
- [math]\displaystyle{ w }[/math] = Out-of-plane displacement of the beam
- [math]\displaystyle{ E }[/math] = modulus of elasticity (Young's Modulus),
- [math]\displaystyle{ I }[/math] = area moment of inertia, and
- [math]\displaystyle{ q }[/math] = distributed load (force per unit length)
Work in Astronomy
Understanding the nature of comets
In addition to his work with classical mechanics, Euler was recognized by Paris Academy Prizes over the course of his career for calculating, with great accuracy, the orbits of comets and other celestial bodies.
Calculating the parallax of the sun
Euler calculated the parallax of the sun, calculating the difference in the apparent position of the object and the actual position of the object. Euler’s calculation of the parallax later led to the development of more accurate longitude tables.
Work in optics
While Newton argued that light was made of particles, Euler argued that light behaved more like waves. In Nova theoria lucis et colorum (1746), Euler argued that diffractions can be more easily argued with the wave theory rather than the previous “pulse theory”. Euler’s wave theory remained the dominant theory about light until the quantum theory of light.
Structural Engineering
Euler also published a formula for calculating the force where the strut would fail that is often used in structural engineering, commonly referred to as Euler's Column Formula.
- [math]\displaystyle{ F=\frac{\pi^2 EI}{(KL)^2} }[/math]
where
- [math]\displaystyle{ F }[/math] = maximum or critical force (vertical load on column),
- [math]\displaystyle{ E }[/math] = modulus of elasticity (Young's Modulus),
- [math]\displaystyle{ I }[/math] = area moment of inertia, and
- [math]\displaystyle{ K L }[/math] is the length of the column.
Fluid Dynamics
Euler in 1757 published a set of equations for flow of an ideal fluid with no viscosity (Inviscid flow) that are now known as the Euler Equations.
- [math]\displaystyle{ \rho\left( \frac{\partial}{\partial t}+{ \mathbf {u}}\cdot\nabla \right){ \mathbf {u}}+\nabla p=0 }[/math]
where
- [math]\displaystyle{ { \mathbf {u}} }[/math] = fluid velocity
- [math]\displaystyle{ p }[/math] = pressure, and
- [math]\displaystyle{ \rho }[/math] = fluid density
See also
Further reading
- Dunham, William. Euler: The Master of Us All. Washington, D.C.: Mathematical Association of America, 1999. Print.
External links
- http://eulerarchive.maa.org/ A digital library dedicated to the work and life of Leonhard Euler
- http://micro.magnet.fsu.edu/optics/timeline/people/euler.html Euler's work on optics
- https://muse.jhu.edu/journals/perspectives_on_science/v016/16.4.pedersen.html Euler's Wave Theory of Light
- http://web.mit.edu/16.unified/www/SPRING/materials/Lectures/M4.7-Unified09.pdf Euler's Column Formula
- http://eulerarchive.maa.org/pages/E088.html Nova theoria lucis et colorum
References
- Weisstein, Eric W. "Euler's Equations of Inviscid Motion." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/EulersEquationsofInviscidMotion.html
- "The Euler-Bernoulli Beam." (2005): 368-401. Center for Aerospace Structures. University of Colorado Boulder, 2005. Web. 1 Dec. 2015.
- Grieve, David J. "Euler Buckling Formula Derivation." Euler Buckling Formula Derivation. Plymouth University, 1 Mar. 2004. Web. 01 Dec. 2015.
- Musielak, Dora. "Euler: Genius Blind Astronomer Mathematician." (n.d.): n. pag. ArXiv.org. Cornell University Library, 28 June 2014. Web. 1 Dec. 2015.