Let f be a continuous function that can be expressed in polar coordinates as a function of θ only; that is, f ( x , y ) = g ( r ) h ( θ ) . where ( x , y ) ∈ D = { ( r , θ ) | R 1 ≤ r ≤ R 2 , θ 1 ≤ θ ≤ θ 2 } , With 0 ≤ R 1 < R 2 , and 0 ≤ θ 1 < θ 2 ≤ 2 π . Show that ∬ D f ( x , y ) d A = [ G ( R 2 ) − F ( R 1 ) ] [ H ( θ 2 ) − H ( θ 1 ) ] . where G and H are antiderivatives of g and h. respectively.
Let f be a continuous function that can be expressed in polar coordinates as a function of θ only; that is, f ( x , y ) = g ( r ) h ( θ ) . where ( x , y ) ∈ D = { ( r , θ ) | R 1 ≤ r ≤ R 2 , θ 1 ≤ θ ≤ θ 2 } , With 0 ≤ R 1 < R 2 , and 0 ≤ θ 1 < θ 2 ≤ 2 π . Show that ∬ D f ( x , y ) d A = [ G ( R 2 ) − F ( R 1 ) ] [ H ( θ 2 ) − H ( θ 1 ) ] . where G and H are antiderivatives of g and h. respectively.
Let f be a continuous function that can be expressed in polar coordinates as a function of
θ
only; that is,
f
(
x
,
y
)
=
g
(
r
)
h
(
θ
)
. where
(
x
,
y
)
∈
D
=
{
(
r
,
θ
)
|
R
1
≤
r
≤
R
2
,
θ
1
≤
θ
≤
θ
2
}
, With
0
≤
R
1
<
R
2
, and
0
≤
θ
1
<
θ
2
≤
2
π
. Show that
∬
D
f
(
x
,
y
)
d
A
=
[
G
(
R
2
)
−
F
(
R
1
)
]
[
H
(
θ
2
)
−
H
(
θ
1
)
]
. where G and H are antiderivatives of g and h. respectively.
Let u(x, y) = e x (x cos y − y sin y). Find all analytic functions f: C → C such that the real part of f(z) is u(x, y).
Does f(z)=z/(sin z)^2 have a pole of order 1 or 2 at z=0?
Let z = f(x, y) be a function on R 2 such that all first-order partial derivatives exist. Let x = r cos θ and y = r sin θ. Use the chain rule to find x ∂f ∂y − y ∂f ∂x in polar coordinates.
Mathematics for Elementary Teachers with Activities (5th Edition)
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