Editing Unit vector (section)

===Cylindrical coordinates===
{{see also|Jacobian matrix}}
The three [[orthogonal]] unit vectors appropriate to cylindrical symmetry are: 
* <math alt="rho-hat">\boldsymbol{\hat{\rho}}</math> (also designated <math alt="e-hat">\mathbf{\hat{e}}</math> or <math alt="s-hat">\boldsymbol{\hat s}</math>), representing the direction along which the distance of the point from the axis of symmetry is measured; 
* <math alt="phi-hat">\boldsymbol{\hat \varphi}</math>, representing the direction of the motion that would be observed if the point were rotating counterclockwise about the [[symmetry axis]];
* <math alt="z-hat">\mathbf{\hat{z}}</math>, representing the direction of the symmetry axis; 
They are related to the Cartesian basis <math alt="x-hat">\hat{x}</math>, <math alt="y-hat">\hat{y}</math>, <math alt="z-hat">\hat{z}</math> by:

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

The vectors <math alt="rho-hat">\boldsymbol{\hat{\rho}}</math> and <math alt="phi-hat">\boldsymbol{\hat \varphi}</math> are functions of <math alt="coordinate phi">\varphi,</math> and are ''not'' constant in direction. When differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. The derivatives with respect to <math>\varphi</math> are:

:<math alt="partial derivative of rho-hat with respect to phi equals minus sine of phi in the x-hat direction plus cosine of phi in the y-hat direction equals phi-hat">\frac{\partial \boldsymbol{\hat{\rho}}} {\partial \varphi} = -\sin \varphi\mathbf{\hat{x}} + \cos \varphi\mathbf{\hat{y}} = \boldsymbol{\hat \varphi}</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 rho-hat">\frac{\partial \boldsymbol{\hat \varphi}} {\partial \varphi} = -\cos \varphi\mathbf{\hat{x}} - \sin \varphi\mathbf{\hat{y}} = -\boldsymbol{\hat{\rho}}</math>
:<math alt="partial derivative of z-hat with respect to phi equals zero"> \frac{\partial \mathbf{\hat{z}}} {\partial \varphi} = \mathbf{0}.</math>