7.2: Coordenadas esféricas
- Page ID
- 113826
Las coordenadas esféricas se definen a partir de coordenadas cartesianas
\[\begin{aligned} r &= \sqrt{x^2+y^2+z^2} \\[4pt] \phi &= \arctan(y/x) \\[4pt] \theta &=\arctan\left(\frac{\sqrt{x^2+y^2}}{z}\right)\end{aligned} \nonumber \]
o alternativamente
\[\begin{aligned} x &= r \cos\phi\sin\theta,\\[4pt] y &= r \sin\phi\sin\theta \\[4pt] z &=r \cos\theta\end{aligned} \nonumber \]
como se indica esquemáticamente en la Fig. \(\PageIndex{1}\).
Usando la regla de la cadena encontramos
\[\begin{aligned} \frac{\partial}{\partial x}{~} &= \frac{\partial r}{\partial x}\frac{\partial}{\partial r}{~} + \frac{\partial \phi}{\partial x}\frac{\partial}{\partial \phi}{~} + \frac{\partial \theta}{\partial x}\frac{\partial}{\partial \theta}{~}\nonumber\\[4pt] &= \frac{x}{r} \frac{\partial}{\partial r}{~}-\frac{y}{x^2+y^2}\frac{\partial}{\partial \phi}{~} +\frac{xz}{r^2\sqrt{x^2+y^2}}\frac{\partial}{\partial \theta}{~} \nonumber\\[4pt] &= \sin\theta\cos\phi\frac{\partial}{\partial r}{~}-\frac{\sin\phi}{r\sin\theta} \frac{\partial}{\partial \phi}{~} +\frac{\cos\phi\cos\theta}{r} \frac{\partial}{\partial \theta}{~},\\[4pt] \frac{\partial}{\partial y}{~} &= \frac{\partial r}{\partial y}\frac{\partial}{\partial r}{~}+\frac{\partial \phi}{\partial y}\frac{\partial}{\partial \phi}{~}+\frac{\partial \theta}{\partial y}\frac{\partial}{\partial \theta}{~}\nonumber\\[4pt] &= \frac{y}{r} \frac{\partial}{\partial r}{~}+\frac{x}{x^2+y^2}\frac{\partial}{\partial \phi}{~} +\frac{yz}{r^2\sqrt{x^2+y^2}}\frac{\partial}{\partial \theta}{~} \nonumber\\[4pt] &= \sin\theta\sin\phi\frac{\partial}{\partial r}{~}+\frac{\cos\phi}{r\sin\theta} \frac{\partial}{\partial \phi}{~} +\frac{\sin\phi\cos\theta}{r} \frac{\partial}{\partial \theta}{~},\\[4pt] \frac{\partial}{\partial z}{~} &= \frac{\partial r}{\partial z}\frac{\partial}{\partial r}{~}+\frac{\partial \phi}{\partial z}\frac{\partial}{\partial \phi}{~}+\frac{\partial \theta}{\partial z}\frac{\partial }{\partial \theta}{~}\nonumber\\[4pt] &= \frac{z}{r} \frac{\partial }{\partial r}{~} -\frac{\sqrt{x^2+y^2}}{r^2}\frac{\partial }{\partial \theta}{~} \nonumber\\[4pt] &= \sin\theta\sin\phi\frac{\partial }{\partial r}{~}-\frac{\sin\theta}{r} \frac{\partial }{\partial \theta}{~}.\\[4pt]\end{aligned} \nonumber \]
una vez más podemos escribir\({\nabla}\) en términos de estas coordenadas.
\[\begin{aligned} {\nabla} &=& \hat{e}_r \frac{\partial}{\partial r}{~}+\hat{e}_\phi \frac{1}{r\sin\theta}\frac{\partial}{\partial \phi}{~} + \hat{e}_\theta \frac{1}{r}\frac{\partial}{\partial \theta}{~}\end{aligned} \nonumber \]donde los vectores unitarios\[\begin{aligned} \hat{e}_r &=& (\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta), \nonumber\\[4pt] \hat{e}_\phi &=& (-\sin\phi,\cos\phi,0), \nonumber\\[4pt] \hat{e}_\theta &=& (\cos\phi\cos\theta,\sin\phi\cos\theta,-\sin\theta).\end{aligned} \nonumber \]
son un conjunto ortonormal. Decimos que las coordenadas esféricas son ortogonales.
Podemos usar esto para evaluar\(\Delta={\nabla}^2\),
\[\Delta = \frac{1}{r^2}\frac{\partial}{\partial r}{~}\left(r^2 \frac{\partial}{\partial r}{~}\right) +\frac{1}{r^2} \frac{1}{\sin\theta} \frac{\partial}{\partial \theta}{~} \left( \sin\theta\frac{\partial}{\partial \theta}{~} \right) + \frac{1}{r^2}\frac{\partial^2}{\partial \phi^2}{~} \nonumber \]
Finalmente, para la integración sobre estas variables necesitamos conocer el volumen del pequeño cuboide contenido entre\(r\) y\(r+\delta r\), y\(\theta\) y\(\theta + \delta\theta\) y\(\phi\) y\(\phi+\delta\phi\).
La longitud de los lados debido a cada uno de estos cambios es\(\delta r\),\(r \delta \theta\) y\(r \sin \theta \delta \theta\), respectivamente (estos son los jacobianos para la conversión de coordenadas cartesianas en coordenadas polares y esféricas, respectivamente). De este modo concluimos que
\[\int_V f(x,y,z) dx dy dz = \int_V f(r,\theta,\phi) r^2\sin\theta \,dr \,d\theta \,d\phi. \nonumber \]