To write Laplace’s equation, Uxx + Uyy =
0, in polar coordinates, we begin with
Ux...
To write Laplace’s equation, Uxx + Uyy =
0, in polar coordinates, we begin with
Ux = (∂U/∂r)(∂r/∂x) + (∂U/∂θ)(∂θ/∂x)
where r = √(x2+y2), θ = arctan (y/x), x =
r cos θ, y = r sin θ. We get
Ux = (cos θ) Ur – (1/r)(sin θ)
Uθ , Uxx = [∂(Ux)/∂r] (∂r/∂x) +
[∂(Ux)/∂θ](∂θ/∂x)
Carry out this computation, as well as that for Uyy.
Since Uxx and Uyy are both expressed in polar
coordinates, their sum gives Laplace...
Given any Cartesian coordinates, (x,y), there are polar
coordinates (?,?)(r,θ) with −?2<?≤?2.−π2<θ≤π2.
Find polar coordinates with...
Given any Cartesian coordinates, (x,y), there are polar
coordinates (?,?)(r,θ) with −?2<?≤?2.−π2<θ≤π2.
Find polar coordinates with −?2<?≤?2−π2<θ≤π2 for the
following Cartesian coordinates:
(a) If (?,?)=(18,−10)(x,y)=(18,−10) then
(?,?)=((r,θ)=( , )),
(b) If (?,?)=(7,8)(x,y)=(7,8) then
(?,?)=((r,θ)=( , )),
(c) If (?,?)=(−10,6)(x,y)=(−10,6) then
(?,?)=((r,θ)=( , )),
(d) If (?,?)=(17,3)(x,y)=(17,3) then
(?,?)=((r,θ)=( , )),
(e) If (?,?)=(−7,−5)(x,y)=(−7,−5) then
(?,?)=((r,θ)=( , )),
(f) If (?,?)=(0,−1)(x,y)=(0,−1) then (?,?)=((r,θ)=( ,))
Write the following numbers in the polar form
re^iθ,
0≤θ<2π
a) 7−7i
r =......8.98............. , θ...
Write the following numbers in the polar form
re^iθ,
0≤θ<2π
a) 7−7i
r =......8.98............. , θ
=.........?..................
Write each of the given numbers in the polar form
re^iθ,
−π<θ≤π
a) (2+2i) / (-sqrt(3)+i)
r =......sqrt(2)........, θ =
.........?..................., .