Jakobijeva matrica je matrica parcijalnih izvoda prvog reda neke funkcije i ima oblik:
J
F
(
M
)
=
(
∂
f
1
∂
x
1
⋯
∂
f
1
∂
x
n
⋮
⋱
⋮
∂
f
m
∂
x
1
⋯
∂
f
m
∂
x
n
)
.
{\displaystyle J_{F}\left(M\right)={\begin{pmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{pmatrix}}.}
Jakobijan je determinanta Jakobijeve matrice. Dobila je naziv po nemačkom matematičaru Karlu Gustavu Jakobiju .
Neka je
F
:
R
n
→
R
m
{\displaystyle F:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}}
n - prostora na Euklidov m -prostor. Takva funkcija ima m komponenti:
F
:
(
x
1
⋮
x
n
)
⟼
(
f
1
(
x
1
,
…
,
x
n
)
⋮
f
m
(
x
1
,
…
,
x
n
)
)
.
{\displaystyle F:{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}\longmapsto {\begin{pmatrix}f_{1}(x_{1},\dots ,x_{n})\\\vdots \\f_{m}(x_{1},\dots ,x_{n})\end{pmatrix}}.}
Tada parcijalni izvodi tih funkcija čine Jakobijevu matricu:
J
F
(
M
)
=
(
∂
f
1
∂
x
1
⋯
∂
f
1
∂
x
n
⋮
⋱
⋮
∂
f
m
∂
x
1
⋯
∂
f
m
∂
x
n
)
.
{\displaystyle J_{F}\left(M\right)={\begin{pmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{pmatrix}}.}
Matrica se označava i kao:
J
F
(
M
)
,
∂
(
f
1
,
…
,
f
m
)
∂
(
x
1
,
…
,
x
n
)
ili
D
(
f
1
,
…
,
f
m
)
D
(
x
1
,
…
,
x
n
)
.
{\displaystyle J_{F}\left(M\right),\qquad {\frac {\partial \left(f_{1},\ldots ,f_{m}\right)}{\partial \left(x_{1},\ldots ,x_{n}\right)}}\qquad {\text{ili}}\qquad {\frac {\mathrm {D} \left(f_{1},\ldots ,f_{m}\right)}{\mathrm {D} \left(x_{1},\ldots ,x_{n}\right)}}.}
U slučaju da su
(
x
1
,
…
,
x
n
)
{\displaystyle (x_{1},\ldots ,x_{n})}
ortogonalne Dekartove koordinate tada matrica odgovara gradijentu komponenti funkcije, tj.
(
∇
F
i
)
{\displaystyle \left(\nabla F_{i}\right)}
U slučaju da je
m
=
n
{\displaystyle m=n}
kvadratna matrica pa ima determinantu , koja se naziva Jakobijeva determinanta ili Jakobijan. Jakobijan se koristi za izračunavanja višestrukih integrala zamenom koordinatnoga sistema
x
~
j
→
x
i
{\displaystyle {\tilde {x}}_{j}\rightarrow x_{i}}
∫
Ω
~
f
(
x
~
1
,
x
~
2
,
…
,
x
~
n
)
d
x
~
1
d
x
~
2
…
d
x
~
n
=
{\displaystyle \int \limits _{\tilde {\Omega }}f({\tilde {x}}_{1},{\tilde {x}}_{2},\dots ,{\tilde {x}}_{n})d{\tilde {x}}_{1}d{\tilde {x}}_{2}\dots d{\tilde {x}}_{n}=}
=
∫
Ω
f
(
x
~
1
,
x
~
2
,
…
,
x
~
n
)
|
D
(
x
~
1
,
x
~
2
,
…
,
x
~
n
)
D
(
x
1
,
x
2
,
…
,
x
n
)
|
d
x
1
d
x
2
…
d
x
n
{\displaystyle =\int \limits _{\Omega }f({\tilde {x}}_{1},{\tilde {x}}_{2},\dots ,{\tilde {x}}_{n}){\bigg |}{\frac {D({\tilde {x}}_{1},{\tilde {x}}_{2},\dots ,{\tilde {x}}_{n})}{D(x_{1},x_{2},\dots ,x_{n})}}{\bigg |}dx_{1}dx_{2}\dots dx_{n}}
uredi
U slučaju transformacije sfernoga koordinatnoga sistema zadanoga sa (r , θ , φ ) na kartezijev koordinatni sistem jednačine transformacije su:
x
1
=
r
sin
θ
cos
ϕ
{\displaystyle x_{1}=r\,\sin \theta \,\cos \phi \,}
x
2
=
r
sin
θ
sin
ϕ
{\displaystyle x_{2}=r\,\sin \theta \,\sin \phi \,}
x
3
=
r
cos
θ
.
{\displaystyle x_{3}=r\,\cos \theta .\,}
Jakobijeva matrica je tada dana sa:
J
F
(
r
,
θ
,
ϕ
)
=
[
∂
x
1
∂
r
∂
x
1
∂
θ
∂
x
1
∂
ϕ
∂
x
2
∂
r
∂
x
2
∂
θ
∂
x
2
∂
ϕ
∂
x
3
∂
r
∂
x
3
∂
θ
∂
x
3
∂
ϕ
]
=
[
sin
θ
cos
ϕ
r
cos
θ
cos
ϕ
−
r
sin
θ
sin
ϕ
sin
θ
sin
ϕ
r
cos
θ
sin
ϕ
r
sin
θ
cos
ϕ
cos
θ
−
r
sin
θ
0
]
.
{\displaystyle J_{F}(r,\theta ,\phi )={\begin{bmatrix}{\dfrac {\partial x_{1}}{\partial r}}&{\dfrac {\partial x_{1}}{\partial \theta }}&{\dfrac {\partial x_{1}}{\partial \phi }}\\[3pt]{\dfrac {\partial x_{2}}{\partial r}}&{\dfrac {\partial x_{2}}{\partial \theta }}&{\dfrac {\partial x_{2}}{\partial \phi }}\\[3pt]{\dfrac {\partial x_{3}}{\partial r}}&{\dfrac {\partial x_{3}}{\partial \theta }}&{\dfrac {\partial x_{3}}{\partial \phi }}\\\end{bmatrix}}={\begin{bmatrix}\sin \theta \,\cos \phi &r\,\cos \theta \,\cos \phi &-r\,\sin \theta \,\sin \phi \\\sin \theta \,\sin \phi &r\,\cos \theta \,\sin \phi &r\,\sin \theta \,\cos \phi \\\cos \theta &-r\,\sin \theta &0\end{bmatrix}}.}
Jakobijan je determinanta te matrice tj:
det
∂
(
x
,
y
,
z
)
∂
(
r
,
θ
,
φ
)
=
r
2
sin
θ
.
{\displaystyle \det {\frac {\partial (x,y,z)}{\partial (r,\theta ,\varphi )}}=r^{2}\sin \theta .}
Odnosno zapreminski element je tada:
d
V
=
|
det
∂
(
x
,
y
,
z
)
∂
(
r
,
θ
,
φ
)
|
d
r
d
θ
d
φ
=
r
2
sin
θ
d
r
d
θ
d
φ
.
{\displaystyle \mathrm {d} V=\left|\det {\frac {\partial (x,y,z)}{\partial (r,\theta ,\varphi )}}\right|\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi =r^{2}\sin \theta \,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi .}
uredi
U polarnom koordinatnom sistemu jednačine transformacije su:
x
=
r
cos
ϕ
;
{\displaystyle x\,=r\,\cos \,\phi ;}
y
=
r
sin
ϕ
.
{\displaystyle y\,=r\,\sin \,\phi .}
Jakobijeva matrica je dana sa:
J
(
r
,
ϕ
)
=
[
∂
x
∂
r
∂
x
∂
ϕ
∂
y
∂
r
∂
y
∂
ϕ
]
=
[
∂
(
r
cos
ϕ
)
∂
r
∂
(
r
cos
ϕ
)
∂
ϕ
∂
(
r
sin
ϕ
)
∂
r
∂
(
r
sin
ϕ
)
∂
ϕ
]
=
[
cos
ϕ
−
r
sin
ϕ
sin
ϕ
r
cos
ϕ
]
{\displaystyle J(r,\phi )={\begin{bmatrix}{\partial x \over \partial r}&{\partial x \over \partial \phi }\\{\partial y \over \partial r}&{\partial y \over \partial \phi }\end{bmatrix}}={\begin{bmatrix}{\partial (r\cos \phi ) \over \partial r}&{\partial (r\cos \phi ) \over \partial \phi }\\{\partial (r\sin \phi ) \over \partial r}&{\partial (r\sin \phi ) \over \partial \phi }\end{bmatrix}}={\begin{bmatrix}\cos \phi &-r\sin \phi \\\sin \phi &r\cos \phi \end{bmatrix}}}
r
{\displaystyle r}
∬
A
d
x
d
y
=
∬
B
r
d
r
d
ϕ
.
{\displaystyle \iint _{A}dx\,dy=\iint _{B}r\,dr\,d\phi .}
Jakobijan Kaplan, W. Advanced Calculus, 3rd ed. Reading, MA: Addison-Wesley 1984.
Herbert Federer: Geometric measure theory. 1. izdanje. Springer, Berlin. 1996. ISBN 978-3-540-60656-7 . 
![{\displaystyle J_{F}(r,\theta ,\phi )={\begin{bmatrix}{\dfrac {\partial x_{1}}{\partial r}}&{\dfrac {\partial x_{1}}{\partial \theta }}&{\dfrac {\partial x_{1}}{\partial \phi }}\\[3pt]{\dfrac {\partial x_{2}}{\partial r}}&{\dfrac {\partial x_{2}}{\partial \theta }}&{\dfrac {\partial x_{2}}{\partial \phi }}\\[3pt]{\dfrac {\partial x_{3}}{\partial r}}&{\dfrac {\partial x_{3}}{\partial \theta }}&{\dfrac {\partial x_{3}}{\partial \phi }}\\\end{bmatrix}}={\begin{bmatrix}\sin \theta \,\cos \phi &r\,\cos \theta \,\cos \phi &-r\,\sin \theta \,\sin \phi \\\sin \theta \,\sin \phi &r\,\cos \theta \,\sin \phi &r\,\sin \theta \,\cos \phi \\\cos \theta &-r\,\sin \theta &0\end{bmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b96346f237957ad53046ca86790d790150852c)
