EGM6321 - Principles of Engineering Analysis 1, Fall 2009 [ edit | edit source ] Mtg 41:
Review for exam 2
P
n
{\displaystyle P_{n}\ }
P
n
−
1
,
P
n
−
2
,
.
.
.
{\displaystyle P_{n-1},P_{n-2},...\ }
{
P
n
}
{\displaystyle \left\{P_{n}\right\}\ }
{
P
n
}
{\displaystyle \left\{P_{n}\right\}\ }
{
Q
n
}
{\displaystyle \left\{Q_{n}\right\}\ }
L
n
=
P
n
{\displaystyle L_{n}=P_{n}\ }
L
n
=
Q
n
{\displaystyle L_{n}=Q_{n}\ }
Δ
(
1
r
)
=
0
{\displaystyle \Delta \ \left({\frac {1}{r}}\right)=0\ }
1
r
=
1
r
P
Q
=
1
r
Q
(
1
−
2
μ
ρ
+
ρ
2
)
−
1
2
{\displaystyle {\frac {1}{r}}={\frac {1}{r_{PQ}}}={\frac {1}{r_{Q}}}\left(1-2\mu \ \rho \ +\rho \ ^{2}\right)^{-{\frac {1}{2}}}\ }
=
1
r
Q
∑
n
P
n
(
μ
)
ρ
n
{\displaystyle ={\frac {1}{r_{Q}}}\sum _{n}P_{n}(\mu \ )\rho \ ^{n}\ }
=
ρ
:=
r
P
r
Q
{\displaystyle =\rho \ :={\frac {r_{P}}{r_{Q}}}\ }
⇒
1
r
=
∑
n
P
n
(
μ
)
r
P
n
r
Q
n
+
1
=
∑
n
H
n
(
μ
,
r
P
,
r
Q
)
{\displaystyle \Rightarrow \ {\frac {1}{r}}=\sum _{n}P_{n}(\mu \ ){\frac {r_{P}^{n}}{r_{Q}^{n+1}}}=\sum _{n}H_{n}\left(\mu \ ,r_{P},r_{Q}\right)\ }
H
n
(
μ
,
r
P
,
r
Q
)
=
H
n
(
x
,
y
,
z
)
{\displaystyle H_{n}\left(\mu \ ,r_{P},r_{Q}\right)=H_{n}\left(x,y,z\right)\ }
Δ
1
r
=
0
=
∑
n
Δ
H
n
(
x
,
y
,
z
)
⇒
Δ
H
n
=
0
{\displaystyle \Delta \ {\frac {1}{r}}=0=\sum _{n}\Delta \ H_{n}(x,y,z)\ \Rightarrow \ \Delta \ H_{n}=0}
P.29-1
P.30-1
ψ
(
r
,
θ
)
=
∑
n
(
A
n
r
n
+
B
n
r
n
+
1
)
(
C
n
P
n
+
D
n
Q
n
)
{\displaystyle \psi \ (r,\theta \ )=\sum _{n}(A_{n}r^{n}+B_{n}r^{n+1})(C_{n}P_{n}+D_{n}Q_{n})\ }
A
n
r
n
+
B
n
r
n
+
1
{\displaystyle A_{n}r^{n}+B_{n}r^{n+1}\ }
P.31-2
C
n
P
n
+
D
n
Q
n
{\displaystyle C_{n}P_{n}+D_{n}Q_{n}\ }
P.32-1
μ
=
sin
θ
{\displaystyle \mu \ =\sin \theta \ \ }
