Parametrization-examples
Polar,cylindrical and spherical coordinates
Faculty of Mathematics and Computer Science University of Warmia and Mazury in Olsztyn
June 10, 2014
Parametrization-examples
Polar coordinates
α (x , y )
x y
R
Let A := {(x , y ); x2+y2≤ R}. Then, to computeR
Af dA we can use polar coordinates
x = r cos α
y = r sin α r ∈ [0, R], α ∈ [0, 2π), dA = rdrd α,
Parametrization-examples
Polar coordinates
α (x , y )
x y
R
Let A := {(x , y ); x2+y2≤ R}. Then, to computeR
Af dA we can use polar coordinates
x = r cos α
y = r sin α r ∈ [0, R], α ∈ [0, 2π), dA = rdrd α, Thus,
Z
A
f (x , y ) dA = Z 2π
0
Z R 0
f (x , y ) r dr d α.
Parametrization-examples
Polar coordinates
In particular,
If A := {(x , y ) ∈ R2; (x − a)2+ (y − b)2=R2}, then we use the following change of coordinates:
x = a + r cos α
y = b + r sin α r ∈ [0, R], α ∈ [0, 2π), dA = r dr d α,
y = b r sin α r ∈ [0, 1], α ∈ [0, 2π), dA = abr dr d α,
Parametrization-examples
Polar coordinates
In particular,
If A := {(x , y ) ∈ R2; (x − a)2+ (y − b)2=R2}, then we use the following change of coordinates:
x = a + r cos α
y = b + r sin α r ∈ [0, R], α ∈ [0, 2π), dA = r dr d α,
If A := {(x , y ) ∈ R2; xa22 +yb22 =1}, then we use the following change of coordinates:
x = a r cos α
y = b r sin α r ∈ [0, 1], α ∈ [0, 2π), dA = abr dr d α,
Parametrization-examples
Cylindrycal coordinates
Let the set A will be a cylinder:
z
R c d
x y
z = z r ∈ [0, R], α ∈ [0, 2π), z ∈ [c, d ] dA = rdrd αdz,
Parametrization-examples
Cylindrycal coordinates
Let the set A will be a cylinder:
z
R c d
x y
then, to computeR
Af dA we can use cylindycal coordinates
x = r cos α y = r sin α
z = z r ∈ [0, R], α ∈ [0, 2π), z ∈ [c, d ] dA = rdrd αdz,
Parametrization-examples
Cylindrycal coordinates
In particular,
If base of a cylider is a circle with a center at (a, b), then we use the following change of coordinates:
x = a + r cos α y = b + r sin α
z = z, r ∈ [0, R], α ∈ [0, 2π), z ∈ [c, d ] dA = rdrd αdz,
y = b r sin α
z = z, r ∈ [0, 1], α ∈ [0, 2π), z ∈ [c, d ] dA = abrdrd αdz,
Parametrization-examples
Cylindrycal coordinates
In particular,
If base of a cylider is a circle with a center at (a, b), then we use the following change of coordinates:
x = a + r cos α y = b + r sin α
z = z, r ∈ [0, R], α ∈ [0, 2π), z ∈ [c, d ] dA = rdrd αdz,
If base of a cylider is an ellipse xa22 +yb22 =1, then we use the following change of coordinates:
x = a r cos α y = b r sin α
z = z, r ∈ [0, 1], α ∈ [0, 2π), z ∈ [c, d ] dA = abrdrd αdz,
R R
β α
r ∈ [0, R]
α ∈ [0, π]
β ∈ [0, 2π)
Let A := {(x , y ); x2+y2+z2≤ R}. Then, to computeR
Af dA we can use spherical coordinates
x = r sin α cos β y = r sin α sin β
z = r cos α dA = r2sin αdrd αd β,
