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Lecture
XVII
Curvilinear Coordinates; Change
of Variables
As
we
saw
in
lecture
16,
in
E
2
we can
use the polar
coordinates
system.
In
this
system,
we
have
a fixed
point
O
and
a fixed
ray
Ox
.
The coordinates
of
a point
P
are
given
by
r
,
the distance from
P
to
O
,
and
θ
the angle made
by
�
�
Ox
to
OP
. We can
OP
and
Ox
,
as
measured
going counterclockwise from
�
�
change
the
system
of
coordiantes
from polar
to Cartesian
through
a system of
equations:
x
=
r
cos
θ,
y
=
r
sin
θ.
This
is
called
the
defining
system
.
To go from the Cartesian
system to the polar
one,
we
use
the
inverse
defining
system
:
r
=
x
2
+
y
2
, θ
= arctan
y
,
x
for (
x, y
) in
the
first quadrant.
Generally,
we can
introduce new nonCartesian
coordinates
u,
v
by writing
x, y
as
functions
of
these new coordiantes:
x
=
g
(
u,
v
)
,
y
=
h
(
u,
v
)
.
These
equations
form
the
defining
system
for
the new coordiantes.
They can
be
summarized
by writting the
position
vector
�
ˆ
j.
R
(
u,
v
) =
x
ˆ
i
+
yj
=
g
(
u,
v
)
ˆ
i
+
h
(
u,
v
)
ˆ
In
working with
curvilinear coordinates,
it is
useful
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 Two '04
 Duorg
 Math, Polar Coordinates, Coordinate system, Polar coordinate system, Coordinate systems, ∂r

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