Orthogonal Curvilinear Coordinates

Let us consider rectangular coordinates \((x,\: y,\: z)\) of any point expressed as function of \((u_1,\: u_2,\: u_3)\) so that

$$\left.\begin{align*} X &= x(u_1,\: u_3,\:u_3)\\ Y &= y(u_1,\: u_3,\:u_3) \\ Z &= z(u_1,\: u_3,\:u_3) \end{align*} \right )\dots (1) $$

On simplification

$$\left.\begin{align*} u_1 &= u_1(x,\: y,\:z)\\ u_2 &= u_2(x,\: y,\:z) \\ u_3 &= u_3(x,\: y,\:z) \end{align*} \right )\dots (2) $$

Here equation (2) associates a unique set of coordinates \(u_1,\: u_2,\: u_3)\) of a given point p with rectangular coordinates \(x,\: y,\: z)\) called curvilinear coordinates of p. let the surfaces \(u_1 = c_1,\:u_2 = c_2,\: u_3 =c_3\) where \(c_1,\: c_2,\: c_3\) are constants, called coordinates surfaces. If the coordinate surfaces intersect at right angles, the curvilinear coordinate system is called orthogonal. Then u₁, u₂ and u₃ coordinate curves of a curvilinear system are analogous to the x, y and z coordinate axes of a rectangular system.

Let \(\vec r = x\hat i+ y\hat j + z\hat k \) be position vector of point P. then the tangent vector to the curve u₁ at P while u₂ and u₃ are constants, is \(\frac {\partial \vec r}{\partial u_1}\) then unit tangent vector along this direction is

\begin{align*} \hat {e_1} &= \frac {\frac {\partial \vec r}{\partial u_1}}{\left |\frac {\partial \vec r}{\partial u_1}\right |} \\ \text {or,}\:h_1\hat {e_1} &=\frac {\partial \vec r}{\partial u_1}\: \text {where}\: h_1 = \left |\frac {\partial \vec r}{\partial u_1}\right | \\ \text {Similarly,}\: \hat {e_2} &= \frac {\frac {\partial \vec r}{\partial u_2}}{\left |\frac {\partial \vec r}{\partial u_2}\right |}\\ \text {or,}\:h_2\hat {e_2} &=\frac {\partial \vec r}{\partial u_2}\: \text {where}\: h_2 = \left |\frac {\partial \vec r}{\partial u_2}\right | \end{align*}

\begin{align*}\text {or,}\:h_3\hat {e_3} &=\frac {\partial \vec r}{\partial u_3}\: \text {where}\: h_2 = \left |\frac {\partial \vec r}{\partial u_3}\right | \end{align*}

When quantities h₁, h₂, h₃ are scalars and are called scale factors.

Again \(\vec r = \vec r (u_1,\: u_2, \: u_3)\)

\begin{align*}d\vec r &= \frac {\partial \vec r}{\partial u_1}du_1 + \frac {\partial \vec r}{\partial u_2}du_2+ \frac {\partial \vec r}{\partial u_3}du_3 \\ &= h_1\: du_1\:\hat {e_1} + h_2\: du_2\: \hat {e_2} + h_3\: du_3\:\hat {e_3} \end{align*}

Now, the differential of arc length ds is determined from

$$ds^2 = d\vec r.d\vec r $$

But for orthogonal systems,

\begin{align*}\hat {e_1}.\hat {e_2} &= \hat {e_2}.\hat {e_3} = \hat {e_3}.\hat {e_1} \\ \therefore ds^2 &= h_1^2du_1^2 +h_2^2du_2^2 + h_3^2du_3^2\end{align*}

Which is arc length.

Now, the volume element for orthogonal curvilinear coordinates system is given by

\begin{align*} dv &=|( h_1\: du_1\:\hat {e_1}).( h_2\: du_2\: \hat {e_2} )\times (h_3\: du_3\:\hat {e_3} )| \\ dv &= h_1\: h_2\: h_3\: du_1\: du_2\: du_3\:\:\:[\because\hat {e_1}.\hat {e_2}\times \hat {e_3} = 1]\end{align*}

Now Gradient, Divergence and Curl in terms of orthogonal curvilinear coordinates

$$\text {Grad}\: \phi = \nabla \phi = \frac {1}{h_1} \frac {\partial \phi }{\partial u_1} \hat {e_1}+ \frac {1}{h_2} \frac {\partial \phi }{\partial u_2} \hat {e_2}+ \frac {1}{h_3} \frac {\partial \phi }{\partial u_3} \hat {e_3} $$

Where \(\phi = \phi (u_1,\: u_2,\:u_3)\) be any scalar.

\( \hat {e_1},\: \hat {e_2},\: \hat {e_3}\) be the unit vector along u₁, u₂, and u₃ respectively.

Divergence

\begin{align*} \text {Let}\: \vec A &= A_1\hat {e_1} + A_2\hat {e_2} + A_3 \hat {e_3} \\ \text {div}\: \vec A &= \nabla . \vec A \\ &= \frac {1}{h_1h_2h_3}\left [ \frac {\partial (A_1h_2h_3)}{\partial u_1}+ \frac {\partial (A_2h_3h_1)}{\partial u_2}+ \frac {\partial (A_3h_1h_2)}{\partial u_3}\right ] \end{align*}

Curl

\begin{align*}\vec {\nabla}.\vec A = \text {curl}\: \vec A = \frac {1}{h_1h_2h_3} \left |\begin{matrix}h_1\hat {e_1} &h_2 \hat {e_2} &h_3\hat {e_3} \\ \frac {\partial}{\partial u_1} &\frac {\partial }{\partial u_2} &\frac {\partial }{\partial u_3} \\ h_1A_1 &h_2A_2 &h_3A_3\end{matrix}\right | \end{align*}

\begin{align*} \delta ^2\phi = \text {laplacian of }\: \phi \\ \frac {1}{h_1h_2h_3} \left [ \frac {\partial }{\partial u_1}\left (\frac {h_2h_3 }{h_1}\frac {\partial \phi }{\partial u_1} \right )+ \frac {\partial }{\partial u_2}\left (\frac {h_3h_1 }{h_2}\frac {\partial \phi }{\partial u_2} \right )+ \frac {\partial }{\partial u_3}\left (\frac {h_1h_2 }{h_3}\frac {\partial \phi }{\partial u_3} \right )\right ]\end{align*}

We can reduce above expression in to the usual expression by replacing \((u_1,\:u_2,\: u_3)\) is replaced by \((x,\: y,\: z)\) and \((\hat {e_1} + \hat {e_2} + \hat {e_3})\) by \(x,\: y,\: z)\).in usual expression h₁ = h₂ = h₃ = 1.

Bibliography

P.B. Adhikari, Bhoj Raj Gautam, Lekha Nath Adhikari. A Textbook of Physics. kathmandu: Sukunda Pustak Bhawan, 2011.

Jha, V. K.; 'Lecture title'; Elementary Vector Analysis; St. Xavier's College, Kathmandu; 2016.