Jakobijan

Jakobijeva matrica je matrica parcijalnih izvoda prvog reda neke funkcije i ima oblik:

$J_{F}\left(M\right)={\begin{pmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{pmatrix}}.$

Jakobijan je determinanta Jakobijeve matrice. Dobila je naziv po nemačkom matematičaru Karlu Gustavu Jakobiju.

Definicija

Neka je $F:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}$ funkcija sa Euklidova n- prostora na Euklidov m-prostor. Takva funkcija ima m komponenti:

$F:{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}\longmapsto {\begin{pmatrix}f_{1}(x_{1},\dots ,x_{n})\\\vdots \\f_{m}(x_{1},\dots ,x_{n})\end{pmatrix}}.$

Tada parcijalni izvodi tih funkcija čine Jakobijevu matricu:

$J_{F}\left(M\right)={\begin{pmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{pmatrix}}.$

Matrica se označava i kao:

$J_{F}\left(M\right),\qquad {\frac {\partial \left(f_{1},\ldots ,f_{m}\right)}{\partial \left(x_{1},\ldots ,x_{n}\right)}}\qquad {\text{ili}}\qquad {\frac {\mathrm {D} \left(f_{1},\ldots ,f_{m}\right)}{\mathrm {D} \left(x_{1},\ldots ,x_{n}\right)}}.$

U slučaju da su $(x_{1},\ldots ,x_{n})$ ortogonalne Dekartove koordinate tada matrica odgovara gradijentu komponenti funkcije, tj. $\left(\nabla F_{i}\right)$.

Jakobijan

U slučaju da je $m=n$ Jakobijeva matrica je kvadratna matrica pa ima determinantu, koja se naziva Jakobijeva determinanta ili Jakobijan. Jakobijan se koristi za izračunavanja višestrukih integrala zamenom koordinatnoga sistema ${\tilde {x}}_{j}\rightarrow x_{i}$ tako da sledi:

$\int \limits _{\tilde {\Omega }}f({\tilde {x}}_{1},{\tilde {x}}_{2},\dots ,{\tilde {x}}_{n})d{\tilde {x}}_{1}d{\tilde {x}}_{2}\dots d{\tilde {x}}_{n}=$

$=\int \limits _{\Omega }f({\tilde {x}}_{1},{\tilde {x}}_{2},\dots ,{\tilde {x}}_{n}){\bigg |}{\frac {D({\tilde {x}}_{1},{\tilde {x}}_{2},\dots ,{\tilde {x}}_{n})}{D(x_{1},x_{2},\dots ,x_{n})}}{\bigg |}dx_{1}dx_{2}\dots dx_{n}$

Sferni koordinatni sistem

U slučaju transformacije sfernoga koordinatnoga sistema zadanoga sa (r, θ, φ) na kartezijev koordinatni sistem jednačine transformacije su:

$x_{1}=r\,\sin \theta \,\cos \phi \,$

$x_{2}=r\,\sin \theta \,\sin \phi \,$

$x_{3}=r\,\cos \theta .\,$

Jakobijeva matrica je tada dana sa:

$J_{F}(r,\theta ,\phi )={\begin{bmatrix}{\dfrac {\partial x_{1}}{\partial r}}&{\dfrac {\partial x_{1}}{\partial \theta }}&{\dfrac {\partial x_{1}}{\partial \phi }}\\[3pt]{\dfrac {\partial x_{2}}{\partial r}}&{\dfrac {\partial x_{2}}{\partial \theta }}&{\dfrac {\partial x_{2}}{\partial \phi }}\\[3pt]{\dfrac {\partial x_{3}}{\partial r}}&{\dfrac {\partial x_{3}}{\partial \theta }}&{\dfrac {\partial x_{3}}{\partial \phi }}\\\end{bmatrix}}={\begin{bmatrix}\sin \theta \,\cos \phi &r\,\cos \theta \,\cos \phi &-r\,\sin \theta \,\sin \phi \\\sin \theta \,\sin \phi &r\,\cos \theta \,\sin \phi &r\,\sin \theta \,\cos \phi \\\cos \theta &-r\,\sin \theta &0\end{bmatrix}}.$

Jakobijan je determinanta te matrice tj:

$\det {\frac {\partial (x,y,z)}{\partial (r,\theta ,\varphi )}}=r^{2}\sin \theta .$

Odnosno zapreminski element je tada:

$\mathrm {d} V=\left|\det {\frac {\partial (x,y,z)}{\partial (r,\theta ,\varphi )}}\right|\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi =r^{2}\sin \theta \,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi .$

Polarni koordinatni sistem

U polarnom koordinatnom sistemu jednačine transformacije su:

$x\,=r\,\cos \,\phi ;$

$y\,=r\,\sin \,\phi .$

Jakobijeva matrica je dana sa: $J(r,\phi )={\begin{bmatrix}{\partial x \over \partial r}&{\partial x \over \partial \phi }\\{\partial y \over \partial r}&{\partial y \over \partial \phi }\end{bmatrix}}={\begin{bmatrix}{\partial (r\cos \phi ) \over \partial r}&{\partial (r\cos \phi ) \over \partial \phi }\\{\partial (r\sin \phi ) \over \partial r}&{\partial (r\sin \phi ) \over \partial \phi }\end{bmatrix}}={\begin{bmatrix}\cos \phi &-r\sin \phi \\\sin \phi &r\cos \phi \end{bmatrix}}$ Jakobijeva determinanta ili Jakobijan je onda jednak $r$. Zbog toga se dvostruki integrali mogu iz kartezijevoga sistema transformisati u polarni sistem na sledeći način:

$\iint _{A}dx\,dy=\iint _{B}r\,dr\,d\phi .$

Literatura

• 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.