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      <text>28.3.1.1 Orthogonal curvilinear coordinates</text>
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      <text><![CDATA[<h3>Orthogonal curvilinear coordinates</h3>
<p>The results shown in the previous section, have been given in terms of the familiar Cartesian \((x,y,z)\) coordinate system. However, other coordinate systems can be used to better describe some physical situations. A set of coordinates \(u=u(x,y,z)\), \(v=v(x,y,z)\) and \(w=w(x,y,z)\) where the directions at any point indicated by \(u\), \(v\) and \(w\) are orthogonal (perpendicular) to each other is referred to as a set of <strong>orthogonal curvilinear coordinates</strong>. With each coordinate is associated a <strong>scale factor</strong> \(h_u\), \(h_v\) or \(h_w\) respectively where \(h_u = \sqrt{\left( \frac{\partial x}{\partial u} \right)^2 + \left( \frac{\partial y}{\partial u} \right)^2 + \left( \frac{\partial z}{\partial u} \right)^2}\) (with similar expressions for \(h_v\) and \(h_w\)). The scale factor gives a measure of how a change in the coordinate changes the position of a point. <br> Two commonly-used sets of orthogonal curvilinear coordinates are <strong>cylindrical polar coordinates</strong> and <strong>spherical polar coordinates</strong>. These are similar to the plane polar coordinates introduced in 17.2 but represent extensions to three dimensions. </p>
<p>This corresponds to plane polar \((\rho,\phi)\) coordinates with an added \(z\)-coordinate directed out of the \(xy\) plane. Normally the variables \(\rho\) and \(\phi\) are used instead of \(r\) and \(\theta\) to give the three coordinates \(\rho\), \(\phi\) and \(z\). A cylinder has equation \(\rho =\) constant. <br> The relationship between the coordinate systems is given by \(\qquad x = \rho \cos \phi\qquad\quad y = \rho \sin \phi\qquad\quad z=z\) (i.e. the same \(z\) is used by the two coordinate systems).<br> See Figure (a).<br> <img src="@@PLUGINFILE@@/37210.svg" alt="Cylindrical polar coordinates " width="591" height="187" class="img-fluid atto_image_button_text-bottom"> </p>
<p>The scale factors \(h_{\rho}\), \(h_{\phi}\) and \(h_z\) are given as follows \(\qquad\displaystyle h_{\rho} = \sqrt{\left( \frac{\partial x}{\partial \rho} \right)^2 + \left( \frac{\partial y}{\partial \rho} \right)^2 + \left( \frac{\partial z}{\partial \rho} \right)^2} = \sqrt{ (\cos \phi)^2 + (\sin \phi)^2 + 0 } = 1\) \(\qquad\displaystyle h_{\phi} = \sqrt{\left( \frac{\partial x}{\partial \phi} \right)^2 + \left( \frac{\partial y}{\partial \phi} \right)^2 + \left( \frac{\partial z}{\partial \phi} \right)^2} = \sqrt{ (- \rho \sin \phi)^2 + (\rho \cos \phi)^2 + 0 } = \rho\) \(\qquad\displaystyle h_{z} = \sqrt{\left( \frac{\partial x}{\partial z} \right)^2 + \left( \frac{\partial y}{\partial z} \right)^2 + \left( \frac{\partial z}{\partial z} \right)^2} =\sqrt{\left( 0^2 + 0^2 + 1^2 \right)} = 1\)</p>

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<h4>Spherical polar coordinates</h4>

<p>In this system a point is referred to by its distance from the origin \(r\) and two angles \(\phi\) and \(\theta\). The angle \(\theta\) is the angle between the positive \(z\)-axis and the line from the origin to the point. The angle \(\phi\) is the angle from the \(x\)-axis to the projection of the point in the \(xy\) plane. A useful analogy is of latitude, longitude and height on Earth.</p>
<ol>
  <li> The variable \(r\) plays the role of height (but height measured above the centre of Earth rather than from the surface).</li>
  <li> The variable \(\theta\) plays the role of latitude but is modified so that \(\theta = 0\) represents the North Pole, \(\theta = 90^{\circ} = \dfrac{\pi}{2}\) represents the equator and \(\theta = 180^{\circ} = \pi\) represents the South Pole.</li>
  <li> The variable \(\phi\) plays the role of longitude.</li>
</ol>
<p>A sphere has equation \(r =\) constant. <br> The relationship between the coordinate systems is given by \(\qquad\displaystyle x = r \sin \theta \cos \phi\qquad y = r \sin \theta \sin \phi\qquad z = r \cos \theta\).<br> See the figure below.<br>
<img src="@@PLUGINFILE@@/3729.svg" alt="Spherical polar coordinates" width="276" height="236" class="img-fluid atto_image_button_text-bottom"><br></p>
<p>The scale factors \(h_r\), \(h_{\theta}\) and \(h_{\phi}\) are given by \(\quad\displaystyle h_{r} = \sqrt{\left( \frac{\partial x}{\partial r} \right)^2 + \left( \frac{\partial y}{\partial r} \right)^2 + \left( \frac{\partial z}{\partial r} \right)^2} = \sqrt{ (\sin \theta \cos \phi)^2 + (\sin \theta \sin \phi)^2 + (\cos \theta)^2 } = 1\) \(\quad\displaystyle h_{\theta} = \sqrt{\left( \frac{\partial x}{\partial \theta} \right)^2 + \left( \frac{\partial y}{\partial \theta} \right)^2 + \left( \frac{\partial z}{\partial \theta} \right)^2}= \sqrt{ (r \cos \theta \cos \phi)^2 + (r \cos \theta \sin \phi )^2 + (- r \sin \theta)^2 } = r\) \(\quad\displaystyle h_{\phi} = \sqrt{\left( \frac{\partial x}{\partial \phi} \right)^2 + \left( \frac{\partial y}{\partial \phi} \right)^2 + \left( \frac{\partial z}{\partial \phi} \right)^2}= \sqrt{ (-r \sin \theta \sin \phi)^2 + (r \sin \theta \cos \phi )^2 + 0 } = r \sin \theta\)</p>]]></text>
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