Editing Unit vector (section)

===Spherical coordinates===

The unit vectors appropriate to spherical symmetry are: <math alt="r-hat">\mathbf{\hat{r}}</math>, the direction in which the radial distance from the origin increases; <math alt="phi-hat">\boldsymbol{\hat{\varphi}}</math>, the direction in which the angle in the ''x''-''y'' plane counterclockwise from the positive ''x''-axis is increasing; and <math alt="theta-hat">\boldsymbol{\hat \theta}</math>, the direction in which the angle from the positive ''z'' axis is increasing. To minimize redundancy of representations, the polar angle <math alt="theta">\theta</math> is usually taken to lie between zero and 180 degrees. It is especially important to note the context of any ordered triplet written in [[spherical coordinates]], as the roles of <math alt="phi-hat">\boldsymbol{\hat \varphi}</math> and <math alt="theta-hat">\boldsymbol{\hat \theta}</math> are often reversed. Here, the American "physics" convention<ref>Tevian Dray and Corinne A. Manogue, Spherical Coordinates, College Math Journal 34, 168-169 (2003).</ref> is used. This leaves the [[azimuthal angle]] <math alt="phi">\varphi</math> defined the same as in cylindrical coordinates. The [[Cartesian coordinate system|Cartesian]] relations are:

:<math alt="r-hat equals sin of theta times cosine of phi in the x-hat direction plus sine of theta times sine of phi in the y-hat direction plus cosine of theta in the z-hat direction">\mathbf{\hat{r}} = \sin \theta \cos \varphi\mathbf{\hat{x}}  + \sin \theta \sin \varphi\mathbf{\hat{y}} + \cos \theta\mathbf{\hat{z}}</math>

:<math alt="theta-hat equals cosine of theta times cosine of phi in the x-hat direction plus cosine of theta times sine of phi in the y-hat direction minus sine of theta in the z-hat direction">\boldsymbol{\hat \theta} = \cos \theta \cos \varphi\mathbf{\hat{x}} + \cos \theta \sin \varphi\mathbf{\hat{y}} - \sin \theta\mathbf{\hat{z}}</math>

:<math alt="phi-hat equals minus sine of phi in the x-hat direction plus cosine of phi in the y-hat direction">\boldsymbol{\hat \varphi} = - \sin \varphi\mathbf{\hat{x}} + \cos \varphi\mathbf{\hat{y}}</math>

The spherical unit vectors depend on both <math alt="phi">\varphi</math> and <math alt="theta">\theta</math>, and hence there are 5 possible non-zero derivatives. For a more complete description, see [[Jacobian matrix and determinant]]. The non-zero derivatives are:

:<math alt="partial derivative of r-hat with respect to phi equals minus sine of theta times sine of phi in the x-hat direction plus sine of theta times cosine of phi in the y-hat direction equals sine of theta in the phi-hat direction">\frac{\partial \mathbf{\hat{r}}} {\partial \varphi} = -\sin \theta \sin \varphi\mathbf{\hat{x}} + \sin \theta \cos \varphi\mathbf{\hat{y}} = \sin \theta\boldsymbol{\hat \varphi}</math>

:<math alt="partial derivative of r-hat with respect to theta equals cosine of theta times cosine of phi in the x-hat direction plus cosine of theta times sine of phi in the y-hat direction minus sine of theta in the z-hat direction equals theta-hat">\frac{\partial \mathbf{\hat{r}}} {\partial \theta} =\cos \theta \cos \varphi\mathbf{\hat{x}} + \cos \theta \sin \varphi\mathbf{\hat{y}} - \sin \theta\mathbf{\hat{z}}= \boldsymbol{\hat \theta}</math>

:<math alt="partial derivative of theta-hat with respect to phi equals minus cosine of theta times sine of phi in the x-hat direction plus cosine of theta times cosine of phi in the y-hat direction equals cosine of theta in the phi-hat direction">\frac{\partial \boldsymbol{\hat{\theta}}} {\partial \varphi} =-\cos \theta \sin \varphi\mathbf{\hat{x}} + \cos \theta \cos \varphi\mathbf{\hat{y}} = \cos \theta\boldsymbol{\hat \varphi}</math>

:<math alt="partial derivative of theta-hat with respect to theta equals minus sine of theta times cosine of phi in the x-hat direction minus sine of theta times sine of phi in the y-hat direction minus cosine of theta in the z-hat direction equals minus r-hat">\frac{\partial \boldsymbol{\hat{\theta}}} {\partial \theta} = -\sin \theta \cos \varphi\mathbf{\hat{x}} - \sin \theta \sin \varphi\mathbf{\hat{y}} - \cos \theta\mathbf{\hat{z}} = -\mathbf{\hat{r}}</math>

:<math alt="partial derivative of phi-hat with respect to phi equals minus cosine of phi in the x-hat direction minus sine of phi in the y-hat direction equals minus sine of theta in the r-hat direction minus cosine of theta in the theta-hat direction">\frac{\partial \boldsymbol{\hat{\varphi}}} {\partial \varphi} = -\cos \varphi\mathbf{\hat{x}} - \sin \varphi\mathbf{\hat{y}} = -\sin \theta\mathbf{\hat{r}} -\cos \theta\boldsymbol{\hat{\theta}}</math>