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[[File:Leonhard Euler.jpg|200px|thumb|right|Leonhard Euler]]
[[File:Leonhard Euler.jpg|200px|thumb|right|Leonhard Euler]]
==Overview==


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 entered University of Basel at the age of 14, with Johann Bernoulli as his mentor.
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 “e”, the concept of a function, summation notation “[Sigma|Σ]”, imaginary unit notation “{{math|''bi''}}”, and popularizing “<ref name="pi">”; however, Euler also made lots of important contributions to physics.  
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 against a distant background due to a perspective shift. When viewed from "Viewpoint A", the object appears to be in front of the blue square. When the viewpoint is changed to "Viewpoint B", the object ''appears'' to have moved in front of the red square.]]
[[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 made on a plate after passing a small circular hole in another plate]]
[[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>


where
where
:<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]],
:<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>L</math> = unsupported length of column,
:<math>K L</math> is the length of the column.
:<math>K</math> = column effective length factor, whose value depends on the conditions of end support of the column, as follows.
::For both ends pinned (hinged, free to rotate), <math>K = 1.0</math>.
::For both ends fixed, <math>K = 0.50</math>.
::For one end fixed and the other end pinned, <math>K = 0.699\ldots</math>.
::For one end fixed and the other end free to move laterally, <math>K = 2.0</math>.
:<math>K L</math> is the effective length of the column.


===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>


=Simple Example=
where
[[File:pathindependence.png]]
:<math>{ \mathbf {u}}</math> = fluid velocity
 
:<math>p</math> = pressure, and
In this example, the electric field is equal to <math> E = \left(E_x, 0, 0\right)</math>. The initial location is A and the final location is C. In order to find the potential difference between A and C, we use <math>dV = V_C - V_A </math>.
:<math>\rho</math> = fluid density
 
Since there are no y and z components of the electric field, the potential difference is <math> dV = -\left(E_x*\left(x_1 - 0\right) + 0*\left(-y_1 - 0\right) + 0*0\right)  = -E_x*x_1</math>
 
[[File:BC.png]]
 
Let's say there is a location B at <math> \left(x_1, 0, 0\right) </math>. Now in order to find the potential difference between A and C, we need to find the potential difference between A and B and then between B and C.
 
The potential difference between A and B is <math>dV = V_B - V_A = -\left(E_x*\left(x_1 - 0\right) + 0*0 + 0*0\right) = -E_x*x_1</math>.
 
The potential difference between B and C is <math>dV = V_C - V_B = -\left(E_x*0 + 0*\left(-y_1 - 0\right) + 0*0\right) = 0</math>.
 
Therefore, the potential difference A and C is <math>V_C - V_A = \left(V_C - V_B\right) + \left(V_B - V_A\right) = E_x*x_1 </math>, which is the same answer that we got when we did not use location B.
 
==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?
 
==History==
 
Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.


== See also ==
== See also ==
 
* [[Daniel Bernoulli]]
Are there related topics or categories in this wiki resource for the curious reader to explore?  How does this topic fit into that context?
* [[Wave-Particle Duality]]
* [[Young's Modulus]]
* [[Multisource Interference: Diffraction]]


===Further reading===
===Further reading===


http://eulerarchive.maa.org/
* Dunham, William. <i>Euler: The Master of Us All</i>. Washington, D.C.: Mathematical Association of America, 1999. Print.
http://micro.magnet.fsu.edu/optics/timeline/people/euler.html
https://muse.jhu.edu/journals/perspectives_on_science/v016/16.4.pedersen.html
http://arxiv.org/ftp/arxiv/papers/1406/1406.7397.pdf
http://blog.mechguru.com/machine-design/how-to-apply-the-euler-bernoulli-beam-theory-for-beam-deflection-calculation/
http://www.colorado.edu/engineering/CAS/courses.d/AVMM.d/AVMM.Ch08.d/AVMM.Ch08.pdf
 
 
Books, Articles or other print media on this topic


===External links===
===External links===


Internet resources on this topic
* 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==


This section contains the the references you used while writing this page
* 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:Which Category did you place this in?]]
[[Category:Notable Scientists]]

Latest revision as of 20:11, 1 December 2015

by Jong Rak Koh

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.

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

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.

Work in optics

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.

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

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.
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