Pagina:Lezioni di analisi matematica.pdf/365

cambiamento di variabili nelle formole, ecc.

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si calcolino le $z'_{u},z'_{v},z'_{u}u$ ecc. È per il teor. del § 83:

$z'_{u}={\frac {\partial z}{\partial u}}={\frac {\partial z}{\partial x}}{\frac {\partial x}{\partial u}}+{\frac {\partial z}{\partial y}}{\frac {\partial y}{\partial u}}$.

$z'_{v}={\frac {\partial z}{\partial v}}={\frac {\partial z}{\partial x}}{\frac {\partial x}{\partial v}}+{\frac {\partial z}{\partial y}}{\frac {\partial y}{\partial v}}$.

Queste formole si possono anche ottenere, notando che:

$dz={\frac {\partial z}{\partial x}}dx+{\frac {\partial z}{\partial y}}dy={\frac {\partial z}{\partial x}}\left[{\frac {\partial x}{\partial u}}du+{\frac {\partial x}{\partial v}}dv\right]+{\frac {\partial z}{\partial y}}\left[{\frac {\partial y}{\partial u}}du+{\frac {\partial y}{\partial v}}dv\right]=$

$=\left({\frac {\partial z}{\partial x}}{\frac {\partial x}{\partial u}}+{\frac {\partial z}{\partial y}}{\frac {\partial y}{\partial u}}\right)du+\left({\frac {\partial z}{\partial x}}{\frac {\partial x}{\partial v}}+{\frac {\partial z}{\partial y}}{\frac {\partial y}{\partial v}}\right)dv$,

e confrontando con:

$dz={\frac {\partial z}{\partial u}}du+{\frac {\partial z}{\partial v}}dv$.

Così con metodo analogo:

$z''_{uu}={\frac {\partial z'_{u}}{\partial u}}={\frac {\partial }{\partial u}}\left[{\frac {\partial z}{\partial x}}{\frac {\partial x}{\partial u}}+{\frac {\partial z}{\partial y}}{\frac {\partial y}{\partial u}}\right]={\frac {\partial z}{\partial x}}{\frac {\partial ^{2}x}{\partial u^{2}}}+{\frac {\partial z}{\partial y}}{\frac {\partial ^{2}y}{\partial u^{2}}}+$

$+{\frac {\partial x}{\partial u}}{\frac {\partial }{\partial u}}\left({\frac {\partial z}{\partial x}}\right)+{\frac {\partial y}{\partial u}}{\frac {\partial }{\partial u}}\left({\frac {\partial z}{\partial y}}\right)=$

${\frac {\partial z}{\partial x}}{\frac {\partial ^{2}}{\partial u^{2}}}+{\frac {\partial z}{\partial y}}{\frac {\partial ^{2}y}{\partial u^{2}}}+{\frac {\partial x}{\partial u}}\left[{\frac {\partial ^{2}z}{\partial x^{2}}}{\frac {\partial x}{\partial u}}+{\frac {\partial ^{2}z}{\partial x\partial y}}{\frac {\partial y}{\partial u}}\right]+$

$+{\frac {\partial y}{\partial u}}\left[{\frac {\partial ^{2}z}{\partial x\partial y}}{\frac {\partial x}{\partial u}}+{\frac {\partial ^{2}z}{\partial y^{2}}}{\frac {\partial y}{\partial u}}\right]$, ecc.

(Si ponga, p. es., $u=\rho ,v=\theta ,x=\rho {\text{cos}}\theta ,y=\rho {\text{sen}}\theta )$.

oss. È facile anche calcolare le $x'_{u},x'_{v},y'_{u},y'_{v}$ quando siano note le $u'_{x},u'_{y},v'_{x},v'_{y}$. Infatti, pensate le $x,y$ come funzioni delle $u,v$ funzioni delle $x,y$ si ha:

$1=x'_{x}=x'_{u}u'_{x}+x'_{v}v'_{x}$,

$0=x'_{y}=x'_{u}u'_{y}+x'_{v}v'_{y}$;

da cui, risolvendo, si ricavano tosto le $x'_{u},x'_{v}$. In modo simile si calcolano le $y'_{u},y'_{v}$.

IV. Talvolta si studia una stessa curva , usando in un primo studio certe coordinate $x,y$, in un altro calcolo altre coordinate $u,v$ [p. es., le coordinate polari $u=\rho ,v=\theta$, dove