# Math Methods 2

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• Created by: Anita
• Created on: 07-05-15 19:23
What is Laplace's equation?
Laplacian of u = 0
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What is Poisson's equation?
Laplacian of u = f
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Define the Newtonian Potential
map from Rm\0 to R, N(x)=1/w2 lnx or 1/(2-m)wm x^2-m
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What is wm in the Newtonian potential?
the surface area of the appropriate unit sphere
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What is the shifted Newtonian potential?
Na(x)=N(x-a) for fixed a
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State Green's integral representation
GIR
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What is the fundamental solution of Laplace's equation?
Laplacian F = delta(x-a)
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When is a function u harmonic?
If Laplacian u = 0
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State the Mean value property for harmonic functions
MVP
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What are the outcomes of the strong maximum principle?
u is constant or u(x)
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What is the outcome of the weak maximum principle?
the maximum of the function is on the boundary
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What is the Dirichlet problem
Laplacian u = f in omega, u=g on the boundary
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What is a Green's function for the Dirichlet problem?
G(x,y)=N(x-y)+v(x,y)
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For a Green's function, what are the conditions on v?
C on omega bar n C2 on omega, laplacian v = 0 on omega, G(x,y)=0 on boundary
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What is the solution of the Dirichlet problem?
SDP
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What is the Heat equation?
du/dt=laplacian u
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Define the heat kernel
map from RmxR to R, K(x,t)=1/(4pit)^m/2 e^-modx^2/4t, 0 if t
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What is the Cauchy problem
du/dt=laplacian u in Rmx[0,infinity) and u=f at t=0 in Rm
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What is the solution to the Cauchy problem?
SCP
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What is the outcome of the maximum principle for the heat equation?
the maximum of u is on the parabolic boundary
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What is the Cauchy Dirichlet problem?
du/dt=laplacian u in omegax(0,T) and u=g on the parabolic boundary
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State the alternative vanishing lemma
AVL
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What is the Euler-Lagrange equation
d/dx(dF/dz(x,u(x),u'(x)))-df/dy(x,u(x),u'(x))=0
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What is an extremal?
A function satisfying the Euler-Lagrange equation
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If F(x,y,z)=F(x,y) what is the EL equation?
df/dy(x,u(x))=0
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If F(x,y,z)=F(x,z) what is the EL equation?
df/dz(x,u'(x))=c constant
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What is Beltrami's identity?
If F(x,y,z)=F(y,z) then E(x)=F(u,u')-u'dF/dz(u,u')=c constant
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## Other cards in this set

### Card 2

#### Front

What is Poisson's equation?

#### Back

Laplacian of u = f

### Card 3

#### Front

Define the Newtonian Potential

### Card 4

#### Front

What is wm in the Newtonian potential?

### Card 5

#### Front

What is the shifted Newtonian potential?