Efter videon följer ett problem som du kan lösa för att testa att du tillgodogjort dig innehållet.
Problem:
Vi kan även uttrycka de polära koordinaterna $r,\theta$ i $x,y$. I det högra halvplanet kan vi t ex skriva\[ \left\{ \begin{array}{cc}r= & \sqrt{x^2+y^2} \\\theta= & \arctan\dfrac{y}{x}.\end{array}\right.\]Beräkna matrisen \[\left( \begin{array}{cc}\dfrac{\partial r}{\partial x} & \dfrac{\partial \theta}{\partial x} \\\dfrac{\partial r}{\partial y} & \dfrac{\partial \theta}{\partial y}\end{array}\right).\]Kontrollera även att denna matris verkligen är inversen till \[ \left( \begin{array}{cc}\dfrac{\partial x}{\partial r} & \dfrac{\partial y}{\partial r} \\ \dfrac{\partial x}{\partial \theta} & \dfrac{\partial y}{\partial \theta} \end{array}\right).\]Derivering ger \[ \begin{array}{cc} \dfrac{\partial r}{\partial x}= & \dfrac{\partial}{\partial x}\sqrt{x^2+y^2}= \dfrac{x}{\sqrt{x^2+y^2}}=\cos\theta, \\ \dfrac{\partial r}{\partial y}= & \dfrac{\partial}{\partial y}\sqrt{x^2+y^2}= \dfrac{y}{\sqrt{x^2+y^2}}=\sin\theta, \\ \dfrac{\partial \theta}{\partial x}= & \dfrac{\partial}{\partial x}\arctan\dfrac{y}{x}= \dfrac{-y}{x^2+y^2}=-\dfrac{\sin\theta}{r}, \\ \dfrac{\partial \theta}{\partial y}= & \dfrac{\partial}{\partial y}\arctan\dfrac{y}{x}= \dfrac{x}{x^2+y^2} =\dfrac{\cos\theta}{r}, \end{array} \]vilket ger matrisen\[ \left( \begin{array}{cc} \cos\theta & -\dfrac{\sin\theta}{r} \\ \sin\theta & \dfrac{\cos\theta}{r} \end{array} \right). \]Enkel matrismultiplikation ger att\[ \left( \begin{array}{cc} \cos\theta & \sin\theta \\ -r\sin\theta & r\cos\theta \end{array} \right) \left( \begin{array}{cc} \cos\theta & -\dfrac{\sin\theta}{r} \\ \sin\theta & \dfrac{\cos\theta}{r} \end{array} \right) = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)\]och\[ \left( \begin{array}{cc} \cos\theta & -\dfrac{\sin\theta}{r} \\ \sin\theta & \dfrac{\cos\theta}{r} \end{array} \right) \left( \begin{array}{cc} \cos\theta & \sin\theta \\ -r\sin\theta & r\cos\theta \end{array} \right) = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right).\]
Maila för handledning.
Hur upplevde du problemet?
Svårighet:
Relevans:
Svårighet:
Relevans: