ABSTRACT
The polar coordinate set (r, θ) and the Cartesian set (x, y) are related by the equations (Figure 4.1a):
x r r x y
y r y x
= = +
= = − cos
sin tan .
θ
θ θ
Referring to the above,
∂ ∂
= = ∂ ∂
= =
∂ ∂
= − = − ∂ ∂
=
r x
x r
r y
y r
x y
r r y x r
cos sin
sin
θ θ
θ θ θ 2 2 =
cos .θ r
Inasmuch as the deflection is a function of r and θ, the chain rule, together with the above relationship, leads to
∂ ∂
= ∂ ∂
− ∂ ∂
w x
w r r
wcos sin .θ θ
θ1 (a)
To evaluate to expression ∂ /∂ ,2 2w x we use Equation a, this time applied to ∂ ∂w / x rather than to w:
∂ ∂
= ∂ ∂
∂ ∂
− ∂∂
∂ ∂
= ∂
1w x r
w x r
w x
cos sinθ θ θ
w r
w r r
w r r∂
− ∂ ∂ ∂
+ ∂ ∂2
2cos sin cos sinθ θ
θ θ θ (b)
+ ∂ ∂
+ ∂ ∂
w r
w rθ
θ θ θ
θsin cos sin .
