Jacobijeva matrika (oznaka
J
{\displaystyle J\,}
J
f
(
x
1
,
… … -->
,
x
n
)
{\displaystyle J_{f}(x_{1},\dots ,x_{n})\,}
matrika , ki jo sestavljajo parcialni odvodi prvega reda vektorja .
Determinanta , ki jo dobimo iz Jakobijeve matrike, se imenuje Jacobijeva determinanta .
Imenujeta se po nemškem matematiku Carlu Gustavu Jacobu Jacobiju (1804 – 1851).
Matrika ima obliko:
J
=
[
∂ ∂ -->
y
1
∂ ∂ -->
x
1
⋯ ⋯ -->
∂ ∂ -->
y
1
∂ ∂ -->
x
n
⋮ ⋮ -->
⋱ ⋱ -->
⋮ ⋮ -->
∂ ∂ -->
y
m
∂ ∂ -->
x
1
⋯ ⋯ -->
∂ ∂ -->
y
m
∂ ∂ -->
x
n
]
{\displaystyle J={\begin{bmatrix}{\dfrac {\partial y_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial y_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial y_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial y_{m}}{\partial x_{n}}}\end{bmatrix}}\,}
V matriki i-ta vrstica odgovarja gradientu i-te komponente funkcije
y
i
{\displaystyle y_{i}\,}
(
∇ ∇ -->
y
i
)
{\displaystyle \left(\nabla y_{i}\right)\,}
Determinanto kvadratne Jacobijeve matrike včasih imenujejo tudi jakobian [ 1]
Jacobijeva matrika
Če je dana preslikava
f
:
R
n
→ → -->
R
m
,
f
=
(
f
1
,
… … -->
,
f
m
)
,
i
=
f
i
(
x
1
,
… … -->
,
x
n
)
,
i
=
1
,
… … -->
,
m
{\displaystyle \mathbf {f} :\mathbb {R} ^{n}\to \mathbb {R} ^{m},\mathbf {f} =(f_{1},\ldots ,f_{m}),_{i}=f_{i}(x_{1},\ldots ,x_{n}),i=1,\ldots ,m\,}
točki
x
{\displaystyle x\,}
parcialni odvodi , potem je dana tudi Jacobijeva matrika razsežnosti
m
× × -->
n
{\displaystyle m\times n\,}
funkcije določa orientacijo tangentne ravnine na funkcijo v dani točki. Tako Jacobijeva matrika posplošuje gradient skalarne funkcije večjega števila spremenljivk.
Jacobijevo matriko označujemo z
J
f
{\displaystyle J_{f}\quad \quad \,}
D
f
{\displaystyle Df\quad \quad \,}
∂ ∂ -->
f
∂ ∂ -->
x
{\displaystyle \textstyle {\frac {\partial f}{\partial x}}\quad \quad \,}
∂ ∂ -->
(
f
1
,
… … -->
,
f
m
)
∂ ∂ -->
(
x
1
,
… … -->
,
x
n
)
{\displaystyle \textstyle {\frac {\partial (f_{1},\ldots ,f_{m})}{\partial (x_{1},\ldots ,x_{n})}}\quad \quad \,}
Primer
Primer 1
Za primer poglejmo pretvorbo sfernih koordinat
(
r
,
θ θ -->
,
ϕ ϕ -->
)
{\displaystyle (r,\theta ,\phi )\,}
kartezični koordinatni sistem
(
x
1
,
x
2
,
x
3
)
{\displaystyle (x_{1},x_{2},x_{3})\,}
f
:
R
+
× × -->
[
0
,
π π -->
]
× × -->
[
0
,
2
π π -->
]
→ → -->
R
3
{\displaystyle f:R^{+}\times [0,\pi ]\times [0,2\pi ]\to R^{3}\,}
x
1
=
r
sin
-->
θ θ -->
cos
-->
ϕ ϕ -->
{\displaystyle x_{1}=r\,\sin \theta \,\cos \phi \,}
x
2
=
r
sin
-->
θ θ -->
sin
-->
ϕ ϕ -->
{\displaystyle x_{2}=r\,\sin \theta \,\sin \phi \,}
x
3
=
r
cos
-->
θ θ -->
.
{\displaystyle x_{3}=r\,\cos \theta .\,}
Jacobijeva matrika je
J
f
(
r
,
θ θ -->
,
ϕ ϕ -->
)
=
[
∂ ∂ -->
x
1
∂ ∂ -->
r
∂ ∂ -->
x
1
∂ ∂ -->
θ θ -->
∂ ∂ -->
x
1
∂ ∂ -->
ϕ ϕ -->
∂ ∂ -->
x
2
∂ ∂ -->
r
∂ ∂ -->
x
2
∂ ∂ -->
θ θ -->
∂ ∂ -->
x
2
∂ ∂ -->
ϕ ϕ -->
∂ ∂ -->
x
3
∂ ∂ -->
r
∂ ∂ -->
x
3
∂ ∂ -->
θ θ -->
∂ ∂ -->
x
3
∂ ∂ -->
ϕ ϕ -->
]
=
[
sin
-->
θ θ -->
cos
-->
ϕ ϕ -->
r
cos
-->
θ θ -->
cos
-->
ϕ ϕ -->
− − -->
r
sin
-->
θ θ -->
sin
-->
ϕ ϕ -->
sin
-->
θ θ -->
sin
-->
ϕ ϕ -->
r
cos
-->
θ θ -->
sin
-->
ϕ ϕ -->
r
sin
-->
θ θ -->
cos
-->
ϕ ϕ -->
cos
-->
θ θ -->
− − -->
r
sin
-->
θ 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}}.}
Determinanta je enaka
r
2
sin
-->
θ θ -->
{\displaystyle r^{2}\sin \theta \,}
Primer 2
Poiščimo Jacobijevo matriko za funkcijo
f
:
R
3
→ → -->
R
4
{\displaystyle f:R^{3}\to R^{4}\,}
y
1
=
x
1
{\displaystyle y_{1}=x_{1}\,}
y
2
=
5
x
3
{\displaystyle y_{2}=5x_{3}\,}
y
3
=
4
x
2
2
− − -->
2
x
3
{\displaystyle y_{3}=4x_{2}^{2}-2x_{3}\,}
y
4
=
x
3
sin
-->
(
x
1
)
{\displaystyle y_{4}=x_{3}\sin(x_{1})\,}
V tem primeru se dobi Jacobijeva matrika
J
f
(
x
1
,
x
2
,
x
3
)
=
[
∂ ∂ -->
y
1
∂ ∂ -->
x
1
∂ ∂ -->
y
1
∂ ∂ -->
x
2
∂ ∂ -->
y
1
∂ ∂ -->
x
3
∂ ∂ -->
y
2
∂ ∂ -->
x
1
∂ ∂ -->
y
2
∂ ∂ -->
x
2
∂ ∂ -->
y
2
∂ ∂ -->
x
3
∂ ∂ -->
y
3
∂ ∂ -->
x
1
∂ ∂ -->
y
3
∂ ∂ -->
x
2
∂ ∂ -->
y
3
∂ ∂ -->
x
3
∂ ∂ -->
y
4
∂ ∂ -->
x
1
∂ ∂ -->
y
4
∂ ∂ -->
x
2
∂ ∂ -->
y
4
∂ ∂ -->
x
3
]
=
[
1
0
0
0
0
5
0
8
x
2
− − -->
2
x
3
cos
-->
(
x
1
)
0
sin
-->
(
x
1
)
]
.
{\displaystyle J_{f}(x_{1},x_{2},x_{3})={\begin{bmatrix}{\dfrac {\partial y_{1}}{\partial x_{1}}}&{\dfrac {\partial y_{1}}{\partial x_{2}}}&{\dfrac {\partial y_{1}}{\partial x_{3}}}\\[3pt]{\dfrac {\partial y_{2}}{\partial x_{1}}}&{\dfrac {\partial y_{2}}{\partial x_{2}}}&{\dfrac {\partial y_{2}}{\partial x_{3}}}\\[3pt]{\dfrac {\partial y_{3}}{\partial x_{1}}}&{\dfrac {\partial y_{3}}{\partial x_{2}}}&{\dfrac {\partial y_{3}}{\partial x_{3}}}\\[3pt]{\dfrac {\partial y_{4}}{\partial x_{1}}}&{\dfrac {\partial y_{4}}{\partial x_{2}}}&{\dfrac {\partial y_{4}}{\partial x_{3}}}\\\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&0&5\\0&8x_{2}&-2\\x_{3}\cos(x_{1})&0&\sin(x_{1})\end{bmatrix}}.}
Iz tega se vidi, da Jacobijeva matrika ni vedno kvadratna.
Jacobijeva determinanta
Kadar je
m
=
n
{\displaystyle m=n\,}
kvadratna in zanjo lahko določimo determinanto . To determinanto imenujemo Jacobijeva determinanta, ki jo včasih imenujemo tudi jakobian .
Primer Jacobijeve determinante
Jacobijeva determinanta za funkcijo
f
:
R
3
→ → -->
R
3
{\displaystyle f:R^{3}\to R^{3}\,}
y
1
=
5
x
2
y
2
=
4
x
1
2
− − -->
2
sin
-->
(
x
2
x
3
)
y
3
=
x
2
x
3
{\displaystyle {\begin{aligned}y_{1}&=5x_{2}\\y_{2}&=4x_{1}^{2}-2\sin(x_{2}x_{3})\\y_{3}&=x_{2}x_{3}\end{aligned}}}
je
|
0
5
0
8
x
1
− − -->
2
x
3
cos
-->
(
x
2
x
3
)
− − -->
2
x
2
cos
-->
(
x
2
x
3
)
0
x
3
x
2
|
=
− − -->
8
x
1
⋅ ⋅ -->
|
5
0
x
3
x
2
|
=
− − -->
40
x
1
x
2
.
{\displaystyle {\begin{vmatrix}0&5&0\\8x_{1}&-2x_{3}\cos(x_{2}x_{3})&-2x_{2}\cos(x_{2}x_{3})\\0&x_{3}&x_{2}\end{vmatrix}}=-8x_{1}\cdot {\begin{vmatrix}5&0\\x_{3}&x_{2}\end{vmatrix}}=-40x_{1}x_{2}.}
Opombe in sklici
Glej tudi
Zunanje povezave
![{\displaystyle f:R^{+}\times [0,\pi ]\times [0,2\pi ]\to R^{3}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e62c3f2e80dff53774abea26329e187d41f9cd2e)
![{\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/e01c07582a1f250ff41ae0b6a86880a36147a558)
![{\displaystyle J_{f}(x_{1},x_{2},x_{3})={\begin{bmatrix}{\dfrac {\partial y_{1}}{\partial x_{1}}}&{\dfrac {\partial y_{1}}{\partial x_{2}}}&{\dfrac {\partial y_{1}}{\partial x_{3}}}\\[3pt]{\dfrac {\partial y_{2}}{\partial x_{1}}}&{\dfrac {\partial y_{2}}{\partial x_{2}}}&{\dfrac {\partial y_{2}}{\partial x_{3}}}\\[3pt]{\dfrac {\partial y_{3}}{\partial x_{1}}}&{\dfrac {\partial y_{3}}{\partial x_{2}}}&{\dfrac {\partial y_{3}}{\partial x_{3}}}\\[3pt]{\dfrac {\partial y_{4}}{\partial x_{1}}}&{\dfrac {\partial y_{4}}{\partial x_{2}}}&{\dfrac {\partial y_{4}}{\partial x_{3}}}\\\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&0&5\\0&8x_{2}&-2\\x_{3}\cos(x_{1})&0&\sin(x_{1})\end{bmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/590d1f0a645a6e5aae3e00cb4cf285e3dcee7f24)
