Mì-fûn

Mì-fûn (Differential Calculus) chhai Sú-ho̍k chûng he tui hâm-sú ke khiu̍k-phu pien-fa-sut ke yit-chúng "sien-sin mèu-shu̍t". Mì-fûn khó-yî khiûn sṳ thi mèu-shu̍t tông hâm-sú chhṳ-pien liòng ke chhí-chhṳ̍t chok chiuk-kèu séu ke kói-pien sṳ̀, hâm-sú ke chhṳ̍t he tsen ngióng kói-pien ke. Tông mêu-sia hâm-sú $\scriptstyle f$ ke chhṳ-pien liòng $\scriptstyle x$ yû yit-ke mì-séu ke kói-pien $\scriptstyle h$ sṳ̀, hâm-sú ke pien-fa cho-tet fûn-kié vì lióng-ke phu-fûn. Yit-ke phu-fûn he sien-sin phu-fûn: chhai yit vì chhìn-khóng hâ, kì chang-pí yî chhṳ-pien liòng ke pien-fa-liòng $\scriptstyle h$ , khó-yî péu-sṳ sṳ̀n $\scriptstyle h$ khi̍p yit-ke lâu $\scriptstyle h$ mò-kôan, chṳ́ lâu hâm-sú $\scriptstyle f$ khi̍p $\scriptstyle x$ yû-kôan ke liòng ke sṳ̀n-chit; chhai kiên kóng-fàm ke chhìn-khóng hâ, kì he yit-ke sien-sin yàng-sa chok-yung chhai $\scriptstyle h$ sông ke chhṳ̍t. Nang yit phu-fûn he pí $\scriptstyle h$ kiên kô kiê ke Mò-khiùng-séu, ya chhiu-he kong chhù yî $\scriptstyle h$ heu yìn-yèn fi tshi yî làng. Tông kói-pien liòng $\scriptstyle h$ hàn séu-sṳ̀, thi-ngi phu-fûn khó-yî fut-lio̍k put-kie, hâm-sú ke pien-fa liòng yok tén yî thi-yit phu-fûn, ya chhiu-he hâm-sú chhai $\scriptstyle x$ chhú ke Mì-fûn, ki chok $\displaystyle f'(x)h$ fe̍t $\displaystyle df_{x}(h)$ .

Li-é

Hâm-sú $f:(x,y)\mapsto \left(x^{2}+y^{2},(1-x^{2}-y^{2})x-y,x-(1-x^{2}-y^{2})y\right)$ he yit-ke chhiùng R² sa to R³ ke hâm-sú. Kì chhai mêu Yit-tiám $(x,y)$ ke Ngâ-khó-pí-kí-chhṳn hhṳn vì:

J_{f}(x,y)={\begin{bmatrix}2x&2y\\1-3x^{2}-y^{2}&-2xy-1\\1+2xy&-1+x^{2}+3y^{2}\end{bmatrix}}

Mì-fûn vì: $df_{(x,y)}:h\mapsto J_{f}(x,y)(h)$ , ya chhiu-he:

df_{(x,y)}:h={\begin{pmatrix}h_{1}\\h_{2}\end{pmatrix}}\mapsto {\begin{bmatrix}2x&2y\\1-3x^{2}-y^{2}&-2xy-1\\1+2xy&-1+x^{2}+3y^{2}\end{bmatrix}}{\begin{pmatrix}h_{1}\\h_{2}\end{pmatrix}}={\begin{pmatrix}2xh_{1}+2yh_{2}\\(1-3x^{2}-y^{2})h_{1}-(2xy+1)h_{2}\\(1+2xy)h_{1}-(1-x^{2}-3y^{2})h_{2}\end{pmatrix}}