where C1+C2=0. Given the boundary conditions in the figure, which coordinate should be assigned to the negative constant (and thus the sinusoidal solutions)?
x
y
C1=C2=0 here
It doesn't matter.
Given the two diff. eq's :
1Xd2Xdx2=C11Yd2Ydy2=C2
where C1+C2=0. Given the boundary conditions in the figure, which coordinate should be assigned to the negative constant (and thus the sinusoidal solutions)?
x
y
C1=C2=0 here
It doesn't matter.
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When does sin(ka)e−ky vanish?
k=0
k=π/(2a)
k=π/a
A and C
A, B, C
Suppose V1(r) and V2(r) are linearly independent functions which both solve Laplace's equation, ∇2V=0.
Does aV1(r)+bV2(r) also solve it (with a and b constants)?
Yes. The Laplacian is a linear operator
No. The uniqueness theorem says this scenario is impossible, there are never two independent solutions!
It is a definite yes or no, but the reasons given above just aren't right!
It depends...
What is the value of ∫2π0sin(2x)sin(3x)dx ?
Zero
π
2π
other
I need resources to do an integral like this!
Separation of Variables (Spherical)
Given ∇2V=0 in Cartesian coords, we separated V(x,y,z)=X(x)Y(y)Z(z). Will this approach work in spherical coordinates, i.e. can we separate V(r,θ,ϕ)=R(r)Θ(θ)Φ(ϕ)?
Sure.
Not quite - the angular components cannot be isolated, e.g., f(r,θ,ϕ)=R(r)Y(θ,ϕ)
It won't work at all because the spherical form of Laplace's Equation has cross terms in it (see the front cover of Griffiths)
Given the two diff. eq's :
1Xd2Xdx2=C11Yd2Ydy2=C2
where C1+C2=0. Given the boundary conditions in the figure, which coordinate should be assigned to the negative constant (and thus the sinusoidal solutions)?
x
y
C1=C2=0 here
It doesn't matter.
CORRECT ANSWER: B