Regular and singular points
Let \(S: \vec{r} = \vec{r}(u,v) \) be a surface and P be a point with position vector
\(\vec{r} = \vec{r}(u,v) \) (i)
Then
\(\vec{r_1} =\frac{\partial \vec{r}}{\partial u} =(\frac{\partial x}{\partial u},\frac{\partial y}{\partial u},\frac{\partial z}{\partial u}) \)
\(\vec{r_2} =\frac{\partial \vec{r}}{\partial v} =(\frac{\partial x}{\partial v},\frac{\partial y}{\partial v},\frac{\partial z}{\partial v}) \)
Now
\( \vec{r_1} \times \vec{r_2} = \begin{pmatrix} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} & \frac{\partial z}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} & \frac{\partial z}{\partial v} \end{pmatrix} \)
If
\( \vec{r_1} \times \vec{r_2} \ne 0 \)
Then
P is called regular point (or ordinary point) on the surface
Otherwise
P is called singularity (singular point) on the surface.
Example 1
If a surface is given by
\(\vec{r} = (u+v, u-v, u.v) \)
Then
\( \vec{r_1}=(1, 1, v) \)
\( \vec{r_2}=(1, -1, u) \)
Now
\( \vec{r_1} \times \vec{r_2} = \begin{pmatrix} \vec{i} & \vec{j} & \vec{k} \\ 1 & 1 & v \\ 1 & -1 & u \end{pmatrix} \)
or
\( \vec{r_1} \times \vec{r_2} = (u-v,v-u,-2) \)
Since
\( \vec{r_1} \times \vec{r_2} \neq 0 \) for all values of \( u \) and \( v \)
There is no singularity in the surface \(\vec{r} = (u+v, u-v, u.v) \).
Example 2
If a surface is given by
\(\vec{r} = (u^3 ,v^3,u^2+v^2) \)
Then
\( \vec{r_1}=( 3 u^2, 0, 2 u) \)
\( \vec{r_2}=(0, 3 v^2, 2 v) \)
Now
\( \vec{r_1} \times \vec{r_2} = \begin{pmatrix} \vec{i} & \vec{j} & \vec{k} \\ 3 u^2 & 0 & 2 u \\ 0 & 3 v^2 & 2 v \end{pmatrix} \)
or
\( \vec{r_1} \times \vec{r_2} = (-6 u v^2, -6 u^2 v, 9 u^2 v^2) \)
or
\( \vec{r_1} \times \vec{r_2} = uv(-6 v, -6 u, 9 uv) \)
Since
\( \vec{r_1} \times \vec{r_2} =0 \) for \( u=0,v=0 \)
There is singularity in the surface \(\vec{r} = (u^3 ,v^3,u^2+v^2) \) at \( u=0,v=0 \)
Essential Singularity
The singularity on the surface that is independent from the particular choice of parametric representation and is due to static nature is called Essential singularity.
For example vertex of the cone is an essential singularity.
Example
If a surface is given by
\(\vec{r} = (u^3 ,v^3,u^2+v^2) \)
Then
\( \vec{r_1}=( 3 u^2, 0, 2 u) \)
\( \vec{r_2}=(0, 3 v^2, 2 v) \)
Now
\( \vec{r_1} \times \vec{r_2} = \begin{pmatrix} \vec{i} & \vec{j} & \vec{k} \\ 3 u^2 & 0 & 2 u \\ 0 & 3 v^2 & 2 v \end{pmatrix} \)
or\( \vec{r_1} \times \vec{r_2} = (-6 u v^2, -6 u^2 v, 9 u^2 v^2) \)
or\( \vec{r_1} \times \vec{r_2} = uv(-6 v, -6 u, 9 uv) \)
Since
\( \vec{r_1} \times \vec{r_2} =0 \) for \( u=0,v=0 \)
There is singularity in the surface \(\vec{r} = (u^3 ,v^3,u^2+v^2) \) at \( u=0,v=0 \)
This singularity at\( u=0,v=0 \) is essential singularity in the surface \(\vec{r} = (u^3 ,v^3,u^2+v^2) \)
Artificial Singularity
The singularity of the surface that is due to some particular choice of parametric representation is called artificial singularity.
For example polar coordinates is an artificial singularity.
Example
If a surface is given by
\(\vec{r} = (u cosv, u sinv, 0) \)
Then
\( \vec{r_1}=( cosv, sinv, 0) \)
\( \vec{r_2}=(-u sinv, u cosv, 0) \)
Now
\( \vec{r_1} \times \vec{r_2} = \begin{pmatrix} \vec{i} & \vec{j} & \vec{k} \\ cosv & sinv & 0 \\ -u sinv & u cosv & 0 \end{pmatrix} \)
or\( \vec{r_1} \times \vec{r_2} = (0,0,u) \)
or\( \vec{r_1} \times \vec{r_2} = u(0,0,1) \)
Since
\( \vec{r_1} \times \vec{r_2} =0 \) for \( u=0 \)
There is singularity in the surface \(\vec{r} = (u cosv, u sinv, 0) \) at \( u=0 \)
This singularity at \( u=0 \) is artificial singularity in the surface \(\vec{r} = (u cosv, u sinv, 0) \)
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