Civil Engineering Reference
In-Depth Information
coordinate
. In the following development, we present the general approach but
leave the details as an end-of-chapter exercise.
The basic relations between the rectangular coordinates
x
,
y
and the cylin-
drical (polar) coordinates
r
,
are
x
=
r
cos
(7.85)
y
=
r
sin
and inversely,
r
2
x
2
y
2
=
+
(7.86)
y
x
tan
=
Per the chain rule of differentiation, we have
∂
T
∂
T
∂
∂
r
x
+
∂
T
∂
∂
∂
x
=
∂
r
∂
x
(7.87)
∂
T
∂
∂
T
∂
∂
r
y
+
∂
T
∂
∂
∂
y
=
r
∂
y
By implicit differentiation of Equation 7.86,
2
r
∂
∂
r
r
x
r
=
x
=
2
x
⇒
x
=
cos
∂
∂
2
r
∂
r
∂
r
y
r
=
y
=
2
y
⇒
y
=
sin
∂
∂
(7.88)
1
sec
2
∂
∂
y
x
2
∂
∂
sin
x
=−
⇒
x
=−
r
1
sec
2
∂
∂
1
x
⇒
∂
∂
cos
y
=
y
=
r
so that Equation 7.87 becomes
∂
T
∂
T
sin
∂
T
∂
x
=
cos
r
−
∂
∂
r
(7.89)
∂
T
∂
T
cos
∂
T
∂
y
=
sin
r
+
∂
∂
r
For the second partial derivatives, we have
∂
∂
∂
2
T
∂
∂
∂
T
∂
∂
T
sin
∂
∂
T
=
=
cos
−
∂
x
2
x
∂
x
r
∂
x
r
∂
x
(7.90)
∂
∂
∂
2
T
∂
∂
∂
∂
T
∂
∂
T
cos
∂
∂
T
=
=
sin
+
y
2
y
∂
y
r
∂
y
r
∂
y
