Cartesian unit vectors in terms of spherical unit vectors

Here’s a way to express the Cartesian unit vectors \hat x, \hat y and \hat z in terms of \hat r, \hat \phi and \hat \theta where \theta is the zenithal angle and \phi is the azimuthal angle.

We have that \begin{cases}x=r\cos\phi\sin\theta \\y=r\sin\phi\sin\theta\\z=r\cos\theta\end{cases}.

Keeping in mind the definition of the gradient operator: \nabla = (\frac{1}{h_r}\frac{\partial}{\partial r}\hat r + \frac{1}{h_\theta}\frac{\partial }{\partial \theta}\hat \theta+\frac{1}{h_\phi}\frac{\partial}{\partial \phi} \hat \phi) where h_{e_i}=|\frac{\partial \vec r}{\partial e_i}|, so that \nabla =(\frac{\partial}{\partial r}\hat r+ \frac{1}{r}\frac{\partial }{\partial \theta} \hat \theta +\frac{1}{r\sin \theta}\frac{\partial}{\partial \phi}\hat \phi).

The Cartesian unit vectors are simply \hat x = \nabla x =\cos\phi\sin \theta \hat r + \cos\phi\cos\theta\hat\theta-\sin\phi\hat\phi,

\hat y = \nabla y =\sin\phi\sin \theta \hat r + \sin\phi\cos\theta\hat\theta+\cos\phi\hat\phi,

\hat z = \nabla z =\cos \theta \hat r-\sin \theta \hat \theta.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s