cours/intégration passage en coordonnées polaires.md
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up:: intégration.changement de variables #s/maths/intégration

D = \mathbb{R}^{2} \setminus (\mathbb{R}^{-} \times \{ 0 \}) \Delta = ]0; +\infty[ \times ]-\pi; \pi[

Soit \begin{align} \varphi : \Delta &\to D \\ \end{align}

\operatorname{Jac}_{\varphi}(r, \theta)= \begin{pmatrix}\frac{ \partial \varphi_1 }{ \partial r } & \frac{ \partial \varphi_1 }{ \partial \theta }\\ \frac{ \partial \varphi_2 }{ \partial r } & \frac{ \partial \varphi_2 }{ \partial \theta } \end{pmatrix}(r, \theta) = \begin{pmatrix}\cos(\theta) & -r \sin(\theta) \\ \sin(\theta) & r \cos(\theta) \end{pmatrix} Et donc : J_{\varphi}(r, \theta) = r le déterminant jacobien de \varphi

$= "![[" + dv.current().file.name + ".svg|700]]"

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f02c8cbbb4493acab7faa5fe0660f93259a76104: \begin{cases}x = r\cos(\theta)\\ y = r\sin(\theta)\end{cases}

160d832c4663c98ae1aa78ddf53ec9eb5a0245fd: (x, y) = \varphi(r, \theta)

b1835908f95b17efc325f47eebe645f1f4b628a9: r

9d12ec40c0a5f567736e3c7d32cbb17a7a7ac4cf: \theta

d852886a1a8c4672c3a01d9e83493fc8943498e6: \varphi(r, \theta) = \begin{pmatrix}r\cos(\theta)\\ r\sin(\theta)\end{pmatrix}

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