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Engineering Equations of the Week is my personal page to document the particular engineering equation that I am using the most that week or holds special significance for the moment. It is also a method for me to improve my familiarity with the Wikimarkup for mathematical formulas .
Week of 23 October 2005 :
Simpson's One-Third Rule
∫
a
b
f
(
x
)
d
x
≈
h
3
[
f
(
x
0
)
+
2
∑
i
=
1
n
/
2
−
1
f
(
x
2
i
)
+
4
∑
j
=
1
n
/
2
f
(
x
2
j
−
1
)
+
f
(
x
n
)
]
{\displaystyle \int _{a}^{b}f(x)dx\approx {\frac {h}{3}}{\bigg [}f(x_{0})+2\sum _{i=1}^{n/2-1}f(x_{2i})+4\sum _{j=1}^{n/2}f(x_{2j-1})+f(x_{n}){\bigg ]}}
Week of 16 October 2005 :
Multidimensional Newton-Raphson method
[
x
i
+
1
y
i
+
1
]
=
[
x
i
y
i
]
−
[
∂
f
∂
x
∂
f
∂
y
∂
g
∂
x
∂
g
∂
y
]
−
1
[
f
(
x
i
,
y
i
)
g
(
x
i
,
y
i
)
]
{\displaystyle {\begin{bmatrix}x_{i+1}\\y_{i+1}\end{bmatrix}}={\begin{bmatrix}x_{i}\\y_{i}\end{bmatrix}}-{\begin{bmatrix}{\partial f \over \partial x}&{\partial f \over \partial y}\\{\partial g \over \partial x}&{\partial g \over \partial y}\end{bmatrix}}^{-1}{\begin{bmatrix}f(x_{i},y_{i})\\g(x_{i},y_{i})\end{bmatrix}}}
Week of 9 October 2005 :
Conservation of momentum for a control volume
∂
∂
t
∫
C
V
V
→
ρ
d
V
+
∫
C
S
V
→
ρ
(
V
→
⋅
n
^
)
d
A
=
Σ
F
→
{\displaystyle {\partial \over \partial t}\int _{CV}^{}{{\vec {V}}\rho }\,d\mathbb {V} +\int _{CS}^{}{{\vec {V}}\rho ({\vec {V}}\cdot {\hat {n}})}\,dA=\Sigma {\vec {F}}}
Week of 2 October 2005 :
Bernoulli's equation
p
+
ρ
v
2
2
+
γ
z
=
c
o
n
s
t
a
n
t
{\displaystyle p+{\rho v^{2} \over 2}+{\gamma z}=\mathrm {constant} }
![{\displaystyle \int _{a}^{b}f(x)dx\approx {\frac {h}{3}}{\bigg [}f(x_{0})+2\sum _{i=1}^{n/2-1}f(x_{2i})+4\sum _{j=1}^{n/2}f(x_{2j-1})+f(x_{n}){\bigg ]}}](/media/api/rest_v1/media/math/render/svg/ac02b51578a2e3cf0ed78c99444470f83f06d525)
