λ
→
0
{\displaystyle \lambda \to 0}
f
=
1
+
x
(
1
+
x
)
2
+
y
2
{\displaystyle f={\frac {1+x}{\sqrt {(1+x)^{2}+y^{2}}}}}
f
x
=
0
{\displaystyle f_{x}=0}
f
y
=
0
{\displaystyle f_{y}=0}
f
x
x
=
0
{\displaystyle f_{xx}=0}
f
x
y
=
0
{\displaystyle f_{xy}=0}
f
y
y
=
−
1
{\displaystyle f_{yy}=-1}
g
=
1
(
1
+
x
)
2
+
y
2
{\displaystyle g={\frac {1}{\sqrt {(1+x)^{2}+y^{2}}}}}
g
x
=
−
1
{\displaystyle g_{x}=-1}
g
y
=
0
{\displaystyle g_{y}=0}
g
x
x
=
−
2
{\displaystyle g_{xx}=-2}
g
x
y
=
0
{\displaystyle g_{xy}=0}
g
y
y
=
−
1
{\displaystyle g_{yy}=-1}
f
=
1
+
x
(
1
+
x
)
2
+
y
2
=
1
−
1
2
y
2
+
O
z
3
{\displaystyle f={\frac {1+x}{\sqrt {(1+x)^{2}+y^{2}}}}=1-{\tfrac {1}{2}}y^{2}+{\mathcal {O}}z^{3}}
f
=
1
+
ξ
(
1
+
ξ
)
2
+
η
2
=
1
−
1
2
η
2
+
O
z
3
{\displaystyle f={\frac {1+\xi }{\sqrt {(1+\xi )^{2}+\eta ^{2}}}}=1-{\tfrac {1}{2}}\eta ^{2}+{\mathcal {O}}z^{3}}
g
=
1
(
1
+
x
)
2
+
y
2
=
1
−
x
−
x
2
?
−
1
2
y
2
+
O
z
3
{\displaystyle g={\frac {1}{\sqrt {(1+x)^{2}+y^{2}}}}=1-x-x^{2}?-{\tfrac {1}{2}}y^{2}+{\mathcal {O}}z^{3}}
g
=
1
(
1
+
ξ
)
2
+
η
2
=
1
−
ξ
−
ξ
2
?
−
1
2
η
2
+
O
z
3
{\displaystyle g={\frac {1}{\sqrt {(1+\xi )^{2}+\eta ^{2}}}}=1-\xi -\xi ^{2}?-{\tfrac {1}{2}}\eta ^{2}+{\mathcal {O}}z^{3}}
m
d
2
r
→
j
d
t
2
=
−
κ
s
(
ℓ
j
,
j
−
1
−
a
)
[
r
→
j
−
r
→
j
−
1
ℓ
j
,
j
−
1
]
+
κ
s
(
ℓ
j
,
j
+
1
−
a
)
[
r
→
j
+
1
−
r
→
j
ℓ
j
+
1
,
j
]
{\displaystyle m{\frac {d^{2}{\vec {r}}_{j}}{dt^{2}}}=-\kappa _{s}\left(\ell _{j,j-1}-a\right)\left[{\frac {{\vec {r}}_{j}-{\vec {r}}_{j-1}}{\ell _{j,j-1}}}\right]+\kappa _{s}\left(\ell _{j,j+1}-a\right)\left[{\frac {{\vec {r}}_{j+1}-{\vec {r}}_{j}}{\ell _{j+1,j}}}\right]}
m
d
2
r
→
j
d
t
2
=
−
κ
s
(
ℓ
j
,
j
−
1
−
a
)
[
r
→
j
−
r
→
j
−
1
ℓ
j
,
j
−
1
]
+
κ
s
(
ℓ
j
,
j
+
1
−
a
)
[
r
→
j
+
1
−
r
→
j
ℓ
j
+
1
,
j
]
{\displaystyle m{\frac {d^{2}{\vec {r}}_{j}}{dt^{2}}}=-\kappa _{s}\left(\ell _{j,j-1}-a\right)\left[{\frac {{\vec {r}}_{j}-{\vec {r}}_{j-1}}{\ell _{j,j-1}}}\right]+\kappa _{s}\left(\ell _{j,j+1}-a\right)\left[{\frac {{\vec {r}}_{j+1}-{\vec {r}}_{j}}{\ell _{j+1,j}}}\right]}
ℓ
j
,
j
−
1
=
|
r
→
j
−
r
→
j
−
1
|
=
(
x
j
−
x
→
j
±
1
)
2
+
(
y
j
−
y
→
j
−
1
)
2
{\displaystyle \ell _{j,j-1}=\left|{\vec {r}}_{j}-{\vec {r}}_{j-1}\right|={\sqrt {(x_{j}-{\vec {x}}_{j\pm 1})^{2}+(y_{j}-{\vec {y}}_{j-1})^{2}}}}
ℓ
→
j
,
j
−
1
=
r
→
j
−
r
→
j
−
1
{\displaystyle {\vec {\ell }}_{j,j-1}={\vec {r}}_{j}-{\vec {r}}_{j-1}}
γ
→
j
=
ξ
j
x
^
+
η
j
y
^
+
ζ
j
z
^
{\displaystyle {\vec {\gamma }}_{j}=\xi _{j}{\hat {x}}\,+\,\eta _{j}{\hat {y}}\,+\,\zeta _{j}{\hat {z}}}
γ
→
j
=
ξ
j
x
^
+
η
j
y
^
{\displaystyle {\vec {\gamma }}_{j}=\xi _{j}{\hat {x}}\,+\,\eta _{j}{\hat {y}}}
F
→
76
=
F
→
on 7
by 6
{\displaystyle {\vec {F}}_{76}={\vec {F}}_{\text{on 7}}^{\,{\text{ by 6}}}}
ℓ
→
76
=
ℓ
x
^
+
γ
→
7
−
γ
→
6
{\displaystyle {\vec {\ell }}_{76}=\ell {\hat {x}}+{\vec {\gamma }}_{7}-{\vec {\gamma }}_{6}}
ℓ
=
1
{\displaystyle \ell =1}
ℓ
→
76
=
x
^
+
γ
→
7
−
γ
→
6
=
(
1
+
Δ
ξ
76
)
x
^
+
(
Δ
η
76
)
y
^
{\displaystyle {\vec {\ell }}_{76}={\hat {x}}+{\vec {\gamma }}_{7}-{\vec {\gamma }}_{6}=(1+\Delta \xi _{76}){\hat {x}}+(\Delta \eta _{76}){\hat {y}}}
Recall that for small
ϵ
{\displaystyle \epsilon }
1
+
ϵ
=
1
+
1
2
ϵ
−
1
8
ϵ
2
+
1
16
ϵ
3
−
5
128
ϵ
4
+
7
256
ϵ
5
−
…
,
,
{\displaystyle {\sqrt {1+\epsilon }}=1+{\tfrac {1}{2}}\epsilon -{\tfrac {1}{8}}\epsilon ^{2}+{\tfrac {1}{16}}\epsilon ^{3}-{\tfrac {5}{128}}\epsilon ^{4}+{\tfrac {7}{256}}\epsilon ^{5}-\ldots ,,}
ℓ
76
=
1
+
2
Δ
ξ
76
+
(
Δ
ξ
76
)
2
+
(
Δ
η
76
)
2
=
1
+
Δ
ξ
76
+
1
2
(
Δ
η
76
)
2
+
O
(
Δ
ξ
)
2
+
O
(
Δ
η
)
4
+
.
.
.
{\displaystyle \ell _{76}={\sqrt {1+2\Delta \xi _{76}+(\Delta \xi _{76})^{2}+(\Delta \eta _{76})^{2}}}=1+\Delta \xi _{76}+{\tfrac {1}{2}}(\Delta \eta _{76})^{2}+{\mathcal {O}}(\Delta \xi )^{2}+{\mathcal {O}}(\Delta \eta )^{4}+...}
Caption text
Header text
Header text
Header text
Example
Example
Example
Example
Example
Example
Example
Example
Example
We create a four-term expression using (4) and (5) . Then we move the terms involving energy density and the Poynting vector to the LHS to obtain:
f
=
m
a
{\displaystyle f=ma}
(10)
e
=
m
c
2
{\displaystyle e=mc^{2}}
(11)
First mention of Rowland[Rowland-2011 1]
First non-Rowland reference [1]
Second mention of roland[Rowland-2011 2]
2nw non-Rowland reference [2]
Second mention of roland[Rowland-2011 3]
↑ Rowland, David R. "The potential energy density in transverse string waves depends critically on longitudinal motion." European journal of physics 32.6 (2011): 1475.
↑ page 1
↑ page 2
![{\displaystyle m{\frac {d^{2}{\vec {r}}_{j}}{dt^{2}}}=-\kappa _{s}\left(\ell _{j,j-1}-a\right)\left[{\frac {{\vec {r}}_{j}-{\vec {r}}_{j-1}}{\ell _{j,j-1}}}\right]+\kappa _{s}\left(\ell _{j,j+1}-a\right)\left[{\frac {{\vec {r}}_{j+1}-{\vec {r}}_{j}}{\ell _{j+1,j}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee1ad98620670ff4c45094eeb691b6df49eb79e7)
