Question

- To write Laplace’s equation, U
_{xx}+ U_{yy}= 0, in polar coordinates, we begin with

U_{x} = (∂U/∂r)(∂r/∂x) + (∂U/∂θ)(∂θ/∂x)

where r = √(x^{2}+y^{2}), θ = arctan (y/x), x =
r cos θ, y = r sin θ. We get

U_{x} = (cos θ) U_{r} – (1/r)(sin θ)
U_{θ} , U_{xx} = [∂(U_{x})/∂r] (∂r/∂x) +
[∂(U_{x})/∂θ](∂θ/∂x)

Carry out this computation, as well as that for U_{yy}.
Since U_{xx} and U_{yy} are both expressed in polar
coordinates, their sum gives Laplace in polar coordinates.

Answer #1

Solve:
uxx + uyy = 0 in {(x,y) st x2 +
y2 < 1 , x > 0, y > 0}
u = 0 on x=0 and y=0
∂u/∂r = 1 on r=1

Your task will be to derive the equations describing the
velocity and acceleration in a polar coordinate
system and a rotating polar vector basis for an object in general
2D motion starting from a general
position vector. Then use these expressions to simplify to the case
of non-uniform circular motion, and
finally uniform circular motion.
Here's the time-dependent position vector in a Cartesian coordinate
system with a Cartesian vector
basis: ⃗r(t)=x (t)
̂
i+y(t)
̂
j where x(t) and y(t)...

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