Math Methods 2 0.0 / 5 ? MathematicsLaplace's EquationHeat EquationCalculus of VariationsUniversityNone Created by: AnitaCreated on: 07-05-15 19:23 What is Laplace's equation? Laplacian of u = 0 1 of 27 What is Poisson's equation? Laplacian of u = f 2 of 27 Define the Newtonian Potential map from Rm\0 to R, N(x)=1/w2 lnx or 1/(2-m)wm x^2-m 3 of 27 What is wm in the Newtonian potential? the surface area of the appropriate unit sphere 4 of 27 What is the shifted Newtonian potential? Na(x)=N(x-a) for fixed a 5 of 27 State Green's integral representation GIR 6 of 27 What is the fundamental solution of Laplace's equation? Laplacian F = delta(x-a) 7 of 27 When is a function u harmonic? If Laplacian u = 0 8 of 27 State the Mean value property for harmonic functions MVP 9 of 27 What are the outcomes of the strong maximum principle? u is constant or u(x) 10 of 27 What is the outcome of the weak maximum principle? the maximum of the function is on the boundary 11 of 27 What is the Dirichlet problem Laplacian u = f in omega, u=g on the boundary 12 of 27 What is a Green's function for the Dirichlet problem? G(x,y)=N(x-y)+v(x,y) 13 of 27 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 14 of 27 What is the solution of the Dirichlet problem? SDP 15 of 27 What is the Heat equation? du/dt=laplacian u 16 of 27 Define the heat kernel map from RmxR to R, K(x,t)=1/(4pit)^m/2 e^-modx^2/4t, 0 if t 17 of 27 What is the Cauchy problem du/dt=laplacian u in Rmx[0,infinity) and u=f at t=0 in Rm 18 of 27 What is the solution to the Cauchy problem? SCP 19 of 27 What is the outcome of the maximum principle for the heat equation? the maximum of u is on the parabolic boundary 20 of 27 What is the Cauchy Dirichlet problem? du/dt=laplacian u in omegax(0,T) and u=g on the parabolic boundary 21 of 27 State the alternative vanishing lemma AVL 22 of 27 What is the Euler-Lagrange equation d/dx(dF/dz(x,u(x),u'(x)))-df/dy(x,u(x),u'(x))=0 23 of 27 What is an extremal? A function satisfying the Euler-Lagrange equation 24 of 27 If F(x,y,z)=F(x,y) what is the EL equation? df/dy(x,u(x))=0 25 of 27 If F(x,y,z)=F(x,z) what is the EL equation? df/dz(x,u'(x))=c constant 26 of 27 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 27 of 27

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