Regra da Cadeia

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Regra da Cadeia

por Pietro di Bernadone Ter 04 Nov 2014, 14:44

Alguém pode me explicar como prosseguir?

Se z = f(x,y), onde x = r.cos θ e y = r.sen θ. Determine:

a) $Regra da Cadeia Gif$

b) $Regra da Cadeia Gif$

c) Mostre que $Regra da Cadeia Gif$

Minha dúvida é na letra c. Alguém pode me ajudar por favor?

Pietro di Bernadone: Grupo
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Re: Regra da Cadeia

por Man Utd Qua 05 Nov 2014, 21:14

$\\\\\\ \frac{\partial z }{\partial r} =\frac{\partial z}{\partial x} \frac{\partial x}{\partial r}+\frac{\partial z}{\partial y} \frac{\partial y}{\partial r} \\\\\\ \frac{\partial z }{\partial r} =\frac{\partial z}{\partial x} \cos \theta+\frac{\partial z}{\partial y} \sin \theta \\\\\\\\ \frac{\partial z }{\partial \theta} =\frac{\partial z}{\partial x} \frac{\partial x}{\partial \theta}+\frac{\partial z}{\partial y} \frac{\partial y}{\partial \theta} \\\\\\ \frac{\partial z }{\partial \theta} =\frac{\partial z}{\partial x} (-r \sin \theta)+\frac{\partial z}{\partial y} \left(r \cos \theta \right)$

$\\\\\\ \left(\frac{\partial z }{\partial r} \right)^2 =\left(\frac{\partial z}{\partial x} \cos \theta+\frac{\partial z}{\partial y} \sin \theta \right)^2 \\\\\\ \left(\frac{\partial z }{\partial \theta} \right )^2=\left(\frac{\partial z}{\partial x} (-r \sin \theta)+\frac{\partial z}{\partial y} \left(r \cos \theta \right) \right)^2 \\\\\\ \left(\frac{\partial z }{\partial r} \right)^2 =\left(\frac{\partial z}{\partial x} \right)^2 \cos^{2} \theta+2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y} \cos \theta \sin \theta+\left(\frac{\partial z}{\partial y}\right)^2 \sin^{2} \theta \\\\\\ \left(\frac{\partial z }{\partial \theta} \right )^2=\left(\frac{\partial z}{\partial x} \right)^2 r^2 \sin^{2} \theta-2r^2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}\cos \theta \sin \theta+\left( \frac{\partial z}{\partial y} \right)^2 r^2 \cos^{2} \theta$

$\\\\\\ \left(\frac{\partial z }{\partial r} \right)^2 +\frac{1}{r^2}\left(\frac{\partial z }{\partial \theta} \right )^2=\left(\frac{\partial z}{\partial x} \right)^2 \cos^{2} \theta+\left(\frac{\partial z}{\partial x} \right)^2 \sin^{2} \theta+2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y} \cos \theta \sin \theta-2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}\cos\theta \sin \theta+\left(\frac{\partial z}{\partial y}\right)^2 \sin^{2} \theta+\left(\frac{\partial z}{\partial y}\right)^2 \cos^{2} \theta$

$\\\\\\ \left(\frac{\partial z }{\partial r} \right)^2 +\frac{1}{r^2}\left(\frac{\partial z }{\partial \theta} \right )^2=\left(\frac{\partial z}{\partial x} \right)^2 \cos^{2} \theta+\left(\frac{\partial z}{\partial x} \right)^2 \sin^{2} \theta+\left(\frac{\partial z}{\partial y}\right)^2 \sin^{2} \theta+\left(\frac{\partial z}{\partial y}\right)^2 \cos^{2} \theta \\\\\\ \left(\frac{\partial z }{\partial r} \right)^2 +\frac{1}{r^2}\left(\frac{\partial z }{\partial \theta} \right )^2=\left(\frac{\partial z}{\partial x} \right)^2 \left( \cos^{2} \theta+ \sin^{2} \theta \right)+\left(\frac{\partial z}{\partial y}\right)^2 \left( \sin^{2} \theta+ \cos^{2} \theta \right) \\\\\\ \boxed{ \boxed{ \boxed{ \left(\frac{\partial z }{\partial r} \right)^2 +\frac{1}{r^2}\left(\frac{\partial z }{\partial \theta} \right )^2=\left(\frac{\partial z}{\partial x} \right)^2 +\left(\frac{\partial z}{\partial y}\right)^2 }}}$