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F16 PH333 3.1.1-3.1.3

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Richard Datwyler

on 15 October 2018

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Transcript of F16 PH333 3.1.1-3.1.3

Big Picture
Laplace's Equation
3.1.1 Introduction
3.1.2 Equation in 1D
3.1.3 Equation in 2D

Electrostatics: Goal, find the electric field - (Coulomb's law)
Problem: Nice geometries only, or bad integrals
Solution: Solve differential form, and use B.C.
Tools: Potentials, Gauss's law
Further: often we want the E field where there are no charges.
Laplace's equation
Its solutions are harmonic functions
Which just means they are second differentiable and continuous
There are some fundamental principles that must be obeyed
averages
extrema
Laplace Equation in 1 D
What are solutions to this equation?
2 unknowns, set by boundary conditions
straight line, no local maxima/minima

smooth and continuous, points on solution are close to their neighbors, specifically they are the average
Let us have a potential in 1D set by B.C. specifying that the potential needs: V(x=2)=0 and V(x=4)=3
Why can't
be a solution to Laplace's equation?
A It doesn't satisfy the B.C.
B It is not smooth and continuous
C There are local maxima/minima
D Each point is not an average of its neighbors
E It IS a solution!
Check the math!
in 1D
Think Straight lines
Set by boundary conditions
Summary 1 D
Each point is an average of its neighbors (this means linear in 1D)
From above, no local maxima or minima
2D
No general solutions - not just a plane
Not specified by 2 or any number of constants
Harmonic Functions (solutions)
Drum head / soap bubble
Key feature remain
2D Laplacian
Each point is still an average of its neighbors
Method of relaxation
Still no local maxima or minima
Visualizations
Note:
'average value'
no local min or max

smoothness
determined by B.C.
"What are harmonic functions? Are they just the result of laplacians, or are they a type of function which laplacians fall into?"
"I'm not quite sure how the relaxation method works."
"I just want some mathematical examples that make geometrical sense"
"Why did the average for the laplacian have a x-a and an x+a? that doesnt really seem like an average?"
Full transcript