Si definisce elemento di volume in
R
k
{\displaystyle \mathbb {R} ^{k}}
k-forma :
d
V
k
=
d
x
1
∧
d
x
2
∧
⋯
∧
d
x
k
.
{\displaystyle \mathrm {d} \mathbf {V} _{k}=\mathrm {d} x_{1}\wedge \mathrm {d} x_{2}\wedge \cdots \wedge \mathrm {d} x_{k}.}
Sia
S
{\displaystyle S}
k
{\displaystyle k}
superficie positivamente orientata in
R
k
{\displaystyle \mathbb {R} ^{k}}
f
{\displaystyle f}
funzione continua definita sull'immagine di
S
{\displaystyle S}
R
{\displaystyle \mathbb {R} }
∫
S
f
(
x
)
d
x
1
∧
d
x
2
∧
⋯
∧
d
x
k
=
∫
S
f
d
V
k
.
{\displaystyle \int _{S}f(\mathbf {x} )\;\mathrm {d} x_{1}\wedge \mathrm {d} x_{2}\wedge \cdots \wedge \mathrm {d} x_{k}=\int _{S}f\;\mathrm {d} \mathbf {V} _{k}.}
Sia
D
⊆
R
k
{\displaystyle D\subseteq \mathbb {R} ^{k}}
S
{\displaystyle S}
S
:
D
→
R
k
{\displaystyle S\colon D\to \mathbb {R} ^{k}}
matrice jacobiana
J
S
{\displaystyle J_{S}}
[ 1]
∫
S
(
D
)
f
(
x
)
d
x
=
∫
D
f
(
S
(
u
)
)
|
J
S
(
u
)
|
d
u
.
{\displaystyle \int _{S(D)}f(\mathbf {x} )\;\mathrm {d} \mathbf {x} =\int _{D}f(S(\mathbf {u} ))\left|J_{S}(\mathbf {u} )\right|\;\mathrm {d} \mathbf {u} .}
Se
f
=
1
{\displaystyle f=1}
Integrale di funzioni su 2-superfici nello spazio
modifica
Sia
S
{\displaystyle S}
R
3
{\displaystyle \mathbb {R} ^{3}}
D
⊆
R
2
{\displaystyle D\subseteq \mathbb {R} ^{2}}
funzioni
x
{\displaystyle x}
y
{\displaystyle y}
z
{\displaystyle z}
u
{\displaystyle u}
v
{\displaystyle v}
S
(
u
,
v
)
=
(
x
(
u
,
v
)
,
y
(
u
,
v
)
,
z
(
u
,
v
)
)
.
{\displaystyle S(u,v)=(x(u,v),y(u,v),z(u,v)).}
Sia
f
:
S
(
D
)
→
R
{\displaystyle f\colon S(D)\to \mathbb {R} }
S
{\displaystyle S}
(
u
,
v
)
∈
D
{\displaystyle (u,v)\in D}
[ 2]
N
(
u
,
v
)
=
∂
(
y
,
z
)
∂
(
u
,
v
)
e
1
+
∂
(
z
,
x
)
∂
(
u
,
v
)
e
2
+
∂
(
x
,
y
)
∂
(
u
,
v
)
e
3
{\displaystyle \mathbf {N} (u,v)={\frac {\partial (y,z)}{\partial (u,v)}}\mathbf {e} _{1}+{\frac {\partial (z,x)}{\partial (u,v)}}\mathbf {e} _{2}+{\frac {\partial (x,y)}{\partial (u,v)}}\mathbf {e} _{3}}
dove i vettori
e
i
{\displaystyle \mathbf {e} _{i}}
base canonica di
R
3
{\displaystyle \mathbb {R} ^{3}}
f
{\displaystyle f}
S
(
D
)
{\displaystyle S(D)}
[ 3]
∫
S
f
d
V
2
=
∫
D
f
(
S
(
u
,
v
)
)
|
N
(
u
,
v
)
|
d
u
d
v
.
{\displaystyle \int _{S}f\;\mathrm {d} \mathbf {V} _{2}=\int _{D}f(S(u,v))|\mathbf {N} (u,v)|\;\mathrm {d} u\;\mathrm {d} v.}
In modo equivalente si scrive anche, notando che il prodotto interno è proprio il vettore normale:
∫
S
f
d
S
=
∬
D
f
(
S
(
u
,
v
)
)
|
∂
S
∂
u
×
∂
S
∂
v
|
d
u
d
v
=
∬
D
f
(
S
(
u
,
v
)
)
|
(
∂
(
y
,
z
)
∂
(
u
,
v
)
,
∂
(
z
,
x
)
∂
(
u
,
v
)
,
∂
(
x
,
y
)
∂
(
u
,
v
)
)
|
d
u
d
v
{\displaystyle \int _{S}f\;\mathrm {d} S=\iint _{D}f(S(u,v))\left|{\partial S \over \partial u}\times {\partial S \over \partial v}\right|\;\mathrm {d} u\;\mathrm {d} v=\iint _{D}f(S(u,v))\left|\left({\frac {\partial (y,z)}{\partial (u,v)}},{\frac {\partial (z,x)}{\partial (u,v)}},{\frac {\partial (x,y)}{\partial (u,v)}}\right)\right|\,\;\mathrm {d} u\;\mathrm {d} v}
dove
N
(
u
,
v
)
=
∂
S
∂
u
×
∂
S
∂
v
=
(
∂
(
y
,
z
)
∂
(
u
,
v
)
,
∂
(
z
,
x
)
∂
(
u
,
v
)
,
∂
(
x
,
y
)
∂
(
u
,
v
)
)
)
{\displaystyle \mathbf {N} (u,v)={\partial S \over \partial u}\times {\partial S \over \partial v}=\left({\frac {\partial (y,z)}{\partial (u,v)}},{\frac {\partial (z,x)}{\partial (u,v)}},{\frac {\partial (x,y)}{\partial (u,v))}}\right)}
è l'elemento di superficie normale a
S
{\displaystyle S}
∂
(
x
i
,
x
j
)
∂
(
u
,
v
)
=
∂
S
x
i
∂
u
∂
S
x
j
∂
v
−
∂
S
x
j
∂
u
∂
S
x
i
∂
v
.
{\displaystyle {\frac {\partial (x_{i},x_{j})}{\partial (u,v)}}={\frac {\partial S_{x_{i}}}{\partial u}}{\frac {\partial S_{x_{j}}}{\partial v}}-{\frac {\partial S_{x_{j}}}{\partial u}}{\frac {\partial S_{x_{i}}}{\partial v}}.}
Se
f
=
1
,
{\displaystyle f=1,}
A
(
S
)
=
∫
D
|
N
(
u
,
v
)
|
d
u
d
v
.
{\displaystyle A(S)=\int _{D}|\mathbf {N} (u,v)|\;\mathrm {d} u\;\mathrm {d} v.}
Sia
S
{\displaystyle S}
R
3
{\displaystyle \mathbb {R} ^{3}}
D
⊆
R
2
{\displaystyle D\subseteq \mathbb {R} ^{2}}
funzioni
x
{\displaystyle x}
y
{\displaystyle y}
z
{\displaystyle z}
u
{\displaystyle u}
v
{\displaystyle v}
S
(
u
,
v
)
=
(
x
(
u
,
v
)
,
y
(
u
,
v
)
,
z
(
u
,
v
)
)
.
{\displaystyle S(u,v)=(x(u,v),y(u,v),z(u,v)).}
Sia
ω
=
ω
x
(
x
,
y
,
z
)
d
y
∧
d
z
+
ω
y
(
x
,
y
,
z
)
d
z
∧
d
x
+
ω
z
(
x
,
y
,
z
)
d
x
∧
d
y
{\displaystyle \omega =\omega _{x}(x,y,z)\;\mathrm {d} y\wedge \;\mathrm {d} z+\omega _{y}(x,y,z)\;\mathrm {d} z\wedge \;\mathrm {d} x+\omega _{z}(x,y,z)\;\mathrm {d} x\wedge \;\mathrm {d} y}
una 2-forma definita su
S
{\displaystyle S}
ω
{\displaystyle \omega }
S
{\displaystyle S}
∫
S
ω
=
∫
D
ω
(
S
(
u
,
v
)
)
[
J
S
(
u
,
v
)
]
d
u
d
v
=
∫
D
[
ω
x
(
S
(
u
,
v
)
)
∂
(
y
,
z
)
∂
(
u
,
v
)
+
ω
y
(
S
(
u
,
v
)
)
∂
(
z
,
x
)
∂
(
u
,
v
)
+
ω
z
(
S
(
u
,
v
)
)
∂
(
x
,
y
)
∂
(
u
,
v
)
]
d
u
d
v
.
{\displaystyle \int _{S}\omega =\int _{D}\omega (S(u,v))[J_{S}(u,v)]\;\mathrm {d} u\;\mathrm {d} v=\int _{D}\left[\omega _{x}(S(u,v)){\frac {\partial (y,z)}{\partial (u,v)}}+\omega _{y}(S(u,v)){\frac {\partial (z,x)}{\partial (u,v)}}+\omega _{z}(S(u,v)){\frac {\partial (x,y)}{\partial (u,v)}}\right]\;\mathrm {d} u\;\mathrm {d} v.}
Interpretando la 2-forma
ω
{\displaystyle \omega }
campo vettoriale
F
=
(
ω
x
,
ω
y
,
ω
z
)
{\displaystyle \mathbf {F} =(\omega _{x},\omega _{y},\omega _{z})}
S
{\displaystyle S}
∫
S
F
⋅
d
S
=
∫
S
(
F
⋅
n
)
d
S
=
∬
D
F
(
S
(
u
,
v
)
)
⋅
n
|
N
(
u
,
v
)
|
d
u
d
v
=
∬
D
F
(
S
(
u
,
v
)
)
⋅
N
(
u
,
v
)
d
u
d
v
{\displaystyle \int _{S}{\mathbf {F} }\cdot \;\mathrm {d} {\mathbf {S} }=\int _{S}({\mathbf {F} }\cdot {\mathbf {n} })\;\mathrm {d} S=\iint _{D}{\mathbf {F} }(S(u,v))\cdot \mathbf {n} \,\,|\mathbf {N} (u,v)|\;\mathrm {d} u\;\mathrm {d} v=\iint _{D}{\mathbf {F} }(S(u,v))\cdot \mathbf {N} (u,v)\;\mathrm {d} u\;\mathrm {d} v}
dove
n
=
N
(
u
,
v
)
|
N
(
u
,
v
)
|
{\displaystyle \mathbf {n} ={\frac {\mathbf {N} (u,v)}{|\mathbf {N} (u,v)|}}}
versore normale alla superficie .
Sia
S
{\displaystyle S}
superficie (chiusa o aperta) analiticamente rappresentata da tre funzioni
x
{\displaystyle x}
y
{\displaystyle y}
z
{\displaystyle z}
ξ
{\displaystyle \xi }
η
{\displaystyle \eta }
x
=
x
(
ξ
,
η
)
,
y
=
y
(
ξ
,
η
)
,
z
=
z
(
ξ
,
η
)
{\displaystyle x=x(\xi ,\eta ),\qquad y=y(\xi ,\eta ),\qquad z=z(\xi ,\eta )}
e sia
f
(
P
)
{\displaystyle f(P)}
funzione continua dei punti
P
(
ξ
,
η
)
{\displaystyle P(\xi ,\eta )}
S
{\displaystyle S}
Δ
s
{\displaystyle \Delta s}
P
(
ξ
,
η
)
{\displaystyle P(\xi ,\eta )}
f
(
P
)
Δ
s
{\displaystyle f(P)\Delta s}
f
(
P
)
{\displaystyle f(P)}
Δ
s
{\displaystyle \Delta s}
∑
Δ
s
=
1
n
f
(
P
)
Δ
s
{\displaystyle \sum _{\Delta s=1}^{n}f(P)\Delta s}
n
{\displaystyle n}
Δ
s
{\displaystyle \Delta s}
limite di tale somma e se è finito, allora esso è l'integrale di superficie della funzione
f
(
P
)
{\displaystyle f(P)}
S
{\displaystyle S}
∫
S
f
(
P
)
⋅
d
s
{\displaystyle \int _{S}f(P)\cdot \mathrm {d} s}
∬
S
f
(
P
)
⋅
d
s
{\displaystyle \iint _{S}f(P)\cdot \mathrm {d} s}
Il suo calcolo effettivo si ottiene mediante un integrale doppio esteso all'area piana
C
{\displaystyle C}
S
{\displaystyle S}
x
y
.
{\displaystyle xy.}
Con lo "spianamento" della superficie
S
{\displaystyle S}
d
s
{\displaystyle \mathrm {d} s}
∬
C
f
(
P
)
⋅
1
+
p
2
+
q
2
⋅
d
C
{\displaystyle \iint _{C}f(P)\cdot {\sqrt {1+p^{2}+q^{2}}}\cdot \mathrm {d} C}
ove
p
=
d
z
/
d
x
{\displaystyle p=\mathrm {d} z/\mathrm {d} x}
q
=
d
z
/
d
y
{\displaystyle q=\mathrm {d} z/\mathrm {d} y}
Walter Rudin, Principi di analisi matematica , Milano, McGraw-Hill, 1991, ISBN 88-386-0647-1 . (EN Volume and Surface Integrals Used in Physics . Cambridge, England: University Press, 1905.
(EN surface integral Enciclopedia Britannica (EN Integrale di superficie MathWorld (EN Surface integral Encyclopaedia of Mathematics Surface Integral — from MathWorld (EN Surface Integral — Theory and exercises PDF math.gatech.edu . 
![{\displaystyle \int _{S}\omega =\int _{D}\omega (S(u,v))[J_{S}(u,v)]\;\mathrm {d} u\;\mathrm {d} v=\int _{D}\left[\omega _{x}(S(u,v)){\frac {\partial (y,z)}{\partial (u,v)}}+\omega _{y}(S(u,v)){\frac {\partial (z,x)}{\partial (u,v)}}+\omega _{z}(S(u,v)){\frac {\partial (x,y)}{\partial (u,v)}}\right]\;\mathrm {d} u\;\mathrm {d} v.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/899decd011fbdd2e14fe7dddc39c475b066c2295)
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