What are Harmonic functions?
Channel: MechanicaLEi
Date: 2018-07-22

Welcome back MechanicaLEi, did you know that the descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion like the swinging of a pendulum? This makes us wonder, what are Harmonic functions? Before we jump in check out the previous part of this series to learn about what Milne-Thomson method is? Now, A harmonic function is a twice continuously differentiable function f which maps the set U into the set R where U is an open subset of R power n that satisfies Laplace's equation that is dou squared f upon dou x 1 squared plus dou squared f upon dou x 2 squared and so on until the sum reaches dou squared f upon dou x n squared and is equals to zero, everywhere on U. This is usually written as delta f equals to zero. One among the many applications of differential equations is to find curves that intersect a given family of curves at right angles. In other words, given a family of curves capital F, we wish to find curve (or curves) Gamma which intersect orthogonally with any member of capital F (whenever they intersect). It is important to note that we are not insisting that Gamma should intersect every member of F but if they intersect, the angle between their tangents, at every point of intersection, is 90 degrees. Such a family of curves Gamma is called "orthogonal trajectories" of the family capital F. That is, at the common point of intersection, the tangents are orthogonal. In case, the family capital F 1 and capital F 2 are identical, we say that the family is self-orthogonal. Hence, we first saw what Harmonic functions are and then went on to see what orthogonal trajectories are?
In the next episode of MechanicaLEi find out what Conformal mapping is?


Attributions:

Doh De Oh by Kevin MacLeod is licensed under a Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/)
Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100255
Artist: http://incompetech.com/

Subtle Library by Fabian Measures (http://freemusicarchive.org/music/Fabian_Measures/) is licensed under a Creative Commons Attribution licence ( https://creativecommons.org/licenses/by/4.0/)
Source: http://freemusicarchive.org/music/Fabian_Measures/Emotion_Music/Subtle_Library

Parabolas with orthogonal trajectories by Ag2gaesh (https://commons.wikimedia.org/wiki/User:Ag2gaeh) is licensed under a Creative Commons Attribution licence ( https://creativecommons.org/licenses/by/4.0/)
Source: https://en.wikipedia.org/wiki/File:Parabel-orth-traj.svg

concentric circles with orthogonal trajectories by Ag2gaesh (https://commons.wikimedia.org/wiki/User:Ag2gaeh) is licensed under a Creative Commons Attribution licence ( https://creativecommons.org/licenses/by/4.0/)
Source: https://en.wikipedia.org/wiki/File:Kreis-orthog-traj.svg