Generado en base a las clases teóricas impartidas en la Escuela Superior de Física y Matemáticas del Instituto Politécnico Nacional - México
Integrales en
R
n
{\displaystyle R^{n}}
Editar Definición : Un rectángulo cerrado en
R
n
{\displaystyle R^{n}}
[
a
1
,
b
1
]
×
[
a
2
,
b
2
]
×
⋯
×
[
a
n
,
b
n
]
{\displaystyle [a_{1},b_{1}]\times [a_{2},b_{2}]\times \dots \times [a_{n},b_{n}]}
con
a
i
,
b
i
∈
R
,
1
≤
i
≤
n
.
{\displaystyle a_{i},b_{i}\in R,1\leq i\leq n.}
Se define también el volumen del rectángulo
S
=
[
a
1
,
b
1
]
×
[
a
2
,
b
2
]
×
⋯
×
[
a
n
,
b
n
]
{\displaystyle S=[a_{1},b_{1}]\times [a_{2},b_{2}]\times \dots \times [a_{n},b_{n}]}
v
o
l
(
S
)
=
∏
i
=
1
n
(
b
i
−
a
i
)
{\displaystyle vol(S)=\prod _{i=1}^{n}(b_{i}-a_{i})}
si
a
i
≤
b
i
∀
1
≤
i
≤
n
{\displaystyle a_{i}\leq b_{i}\forall 1\leq i\leq n}
v
o
l
(
S
)
=
0
{\displaystyle vol(S)=0}
S
=
∅
{\displaystyle S=\emptyset }
Definición : Una partición del rectángulo
S
{\displaystyle S}
P
=
(
P
1
,
P
2
,
…
,
P
n
)
=
P
1
×
P
2
×
⋯
×
P
n
{\displaystyle P=(P_{1},P_{2},\dots ,P_{n})=P_{1}\times P_{2}\times \dots \times P_{n}}
P
j
,
1
≤
j
≤
n
{\displaystyle P_{j},1\leq j\leq n}
[
a
j
,
b
j
]
{\displaystyle [a_{j},b_{j}]}
![{\displaystyle [a_{1},b_{1}]\times [a_{2},b_{2}]\times \dots \times [a_{n},b_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd0ffafe84f97064245bc293cc9dd9fecdfa5afc)
![{\displaystyle S=[a_{1},b_{1}]\times [a_{2},b_{2}]\times \dots \times [a_{n},b_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a917e609141368af4c0a79650cc39dc07c3d6460)
![{\displaystyle [a_{j},b_{j}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dff7f586fc93d2a8f843e8addec31218aafdac74)