Mì-fûn

Hâm-sú chhai Yit-tiám ke Mì-fûn. Khì-chûng fùng-sien phu-fûn he Mì-fûn liòng $dy$, yì kâ sông fôi-sien phu-fûn heu he sṳ̍t-chi ke kói-pien liòng $\mathrm{\Delta }y$

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 $f$ ke chhṳ-pien liòng $x$ yû yit-ke mì-séu ke kói-pien $h$ 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 $h$, khó-yî péu-sṳ sṳ̀n $h$ khi̍p yit-ke lâu $h$ mò-kôan, chṳ́ lâu hâm-sú $f$ khi̍p $x$ 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 $h$ sông ke chhṳ̍t. Nang yit phu-fûn he pí $h$ kiên kô kiê ke Mò-khiùng-séu, ya chhiu-he kong chhù yî $h$ heu yìn-yèn fi tshi yî làng. Tông kói-pien liòng $h$ 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 $x$ chhú ke Mì-fûn, ki chok $f^{\prime }(x)h$ fe̍t $d{f}_x(h)$.

Li-é

Hâm-sú $f:(x,y)\mapsto (x^2+y^2,(1-x^2-y^2)x-y,x-(1-x^2-y^2)y)$ 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:

$J_f(x,y)=\left[\begin{array}{cc}2x& 2y\\ 1-3x^2-y^2& -2xy-1\\ 1+2xy& -1+x^2+3y^2\end{array}\right]$

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

$d{f}_{(x,y)}:h=\left(\begin{array}{c}{h}_1\\ h_2\end{array}\right)\mapsto \left[\begin{array}{cc}2x& 2y\\ 1-3x^2-y^2& -2xy-1\\ 1+2xy& -1+x^2+3y^2\end{array}\right]\left(\begin{array}{c}{h}_1\\ h_2\end{array}\right)=\left(\begin{array}{c}2x{h}_1+2y{h}_2\\ (1-3x^2-y^2)h_1-(2xy+1)h_2\\ (1+2xy)h_1-(1-x^2-3y^2)h_2\end{array}\right)$