Jakobijeva matrica je matrica parcijalnih izvoda prvog reda neke funkcije i ima oblik:
![{\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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba3e9291eae92a2100e79e11a72eaaca2f5c3770)
Jakobijan je determinanta Jakobijeve matrice. Dobila je naziv po nemačkom matematičaru Karlu Gustavu Jakobiju.
Neka je
funkcija sa Euklidova n- prostora na Euklidov m-prostor. Takva funkcija ima m komponenti:
![{\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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90e58f9a09d3ab8feddfd0a7a67915bd8c152bd4)
Tada parcijalni izvodi tih funkcija čine Jakobijevu matricu:
![{\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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba3e9291eae92a2100e79e11a72eaaca2f5c3770)
Matrica se označava i kao:
![{\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)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f9f2e6c9d882b2e047b2a80fdb9805c2039863)
U slučaju da su
ortogonalne Dekartove koordinate tada matrica odgovara gradijentu komponenti funkcije, tj.
.
U slučaju da je
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
tako da sledi:
![{\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}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2364469fe46dfdd4a6886f10362f8a5b9407e82b)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8637cfa7dc5c57c9f211d2f182bed3469af658e5)
U slučaju transformacije sfernoga koordinatnoga sistema zadanoga sa (r, θ, φ) na kartezijev koordinatni sistem jednačine transformacije su:
![{\displaystyle x_{1}=r\,\sin \theta \,\cos \phi \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2e11b36098d514591fd67d4849e7e9f08817e7)
![{\displaystyle x_{2}=r\,\sin \theta \,\sin \phi \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98ff79a1967415422363ab44d26546f1d326b4b8)
![{\displaystyle x_{3}=r\,\cos \theta .\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd64555856e2c0055b0b3d8c2b56a392ae3e3e6c)
Jakobijeva matrica je tada dana sa:
![{\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)
Jakobijan je determinanta te matrice tj:
![{\displaystyle \det {\frac {\partial (x,y,z)}{\partial (r,\theta ,\varphi )}}=r^{2}\sin \theta .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a5f2641b7e9afc4c9dd56c82765964826c9e860)
Odnosno zapreminski element je tada:
![{\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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/911364eca6b9ee3ac59dbcaf8ad19752985fee73)
U polarnom koordinatnom sistemu jednačine transformacije su:
![{\displaystyle x\,=r\,\cos \,\phi ;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7496670efc1b880339098c2b57046b1c838bc213)
![{\displaystyle y\,=r\,\sin \,\phi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f33d8f62cfcc685cadf4ede04b2c8c8a9b6ace3b)
Jakobijeva matrica je dana sa:
Jakobijeva determinanta ili Jakobijan je onda jednak
. Zbog toga se dvostruki integrali mogu iz kartezijevoga sistema transformisati u polarni sistem na sledeći način:
![{\displaystyle \iint _{A}dx\,dy=\iint _{B}r\,dr\,d\phi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e61372c0126887bf7a0b5b0c28b5615f4bac013e)
- 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.