We recall that, space curve is uniquely determined by two local invariant quantities: curvature and torsion.Similarly, surface is uniquely determined by two local invariant quantities: first and second fundamental form of surface
First Fundamental Form of Surface
Let \( S:\vec{r}=\vec{r}( u,v ) \) be a surface, then, quadratic differential form in \( du,dv \) given by
\( I:Edu^2+2Fdudv+Gdv^2 \) where \( E=\vec{r_1}^2,F=\vec{r}_1.\vec{r}_2,G=\vec{r}_2^2 \)
is called first fundamental form of the surface. The coefficients \( E,F,G\) are called the first fundamental coefficients or magnitude of first order.
Geometry of First Fundamental Form of Surface
Let \( S:\vec{r}=\vec{r}( u,v ) \) be a surfacein which \( P \) and \( Q \) are two neighboring points with
\( \overrightarrow{OP}=\vec{r} \) and \( \overrightarrow{OQ}=\vec{r}+d\vec{r}\)
Then
\( \overrightarrow{PQ}=\overrightarrow{OQ}\overrightarrow{OP} \)
or \( d\vec{r}=\vec{r}( u+du,v+dv )\vec{r}( u,v) \)
or
\( d\vec{r}=\vec{r}( u,v )+( \vec{r}_1du+\vec{r}_2dv )+\frac{1}{2!}(\vec{r}_1du+\vec{r}_2dv )^2+...\vec{r}( u,v ) \)
or
\( d\vec{r}=( \vec{r}_1du+\vec{r}_2dv )+\frac1{2!}{{( \vec{r}_1du+\vec{r}_2dv )}^2}+\ldots \)
The first order approximation is
\(dr=( \vec{r}_1du+\vec{r}_2dv ) \)
Squaring on both sides, we get
\( ds^2=(\vec{r}_1du+\vec{r}_2dv )^2 \)
or
\( ds^2=\vec{r}_1^2du^2+2\vec{r}_1\vec{r}_2dudv+\vec{r}_2^2dv^2 \)
or
\( ds^2=Edu^2+2Fdudv+Gdv^2\) where \( E=\vec{r}_1^2,F=\vec{r}_1.\vec{r}_2,G=\vec{r}_2^2 \)
 For \( u\) curve \( ds^2=Edu^2 \)
 For \( v \) curve \( ds^2=Gdv^2 \)
By the ordinary property of vector operation,
\( ( \vec{r}_1\times \vec{r}_2)^2=\vec{r}_1^2. \vec{r}_2^2( \vec{r}_1.\vec{r}_2 )^2 \)
or \( (H \vec{N})^2=E.GF^2 \)
or \(H^2=E.GF^2 \), which is always positive.
Property 1 : First Fundamental Form is Positive
The first fundamental form of the surface is positive definite in \( du \) and \( dv \).
Proof
Let \(S:\vec{r}=\vec{r}( u,v ) \) be a surface, then
\( H^2=E.GF^2 \) is always positive
Now, assume that \( E >0 \), then
\(I=Edu^2+2Fdudv+Gdv^2 \)
or \(I=\frac{1}{E}( E^2du^2+2EFdudv+EGdv^2 )\)
or \(I=\frac{1}{E}[ ( Edu+Fdv )^2+( EGF^2 )dv^2 ]\)
or \(I=\frac{1}{E}[ ( Edu+Fdv )^2+H^2dv^2 ]\)
Here, \(I \) has two possibilities, one is \( I=0 \) and other is \( I>0 \)
If possible, suppose that
\( I=0 \)
Then, we must have
\( \frac{1}{E}[ ( Edu+Fdv )^2+H^2dv^2 ]=0 \)
or \( ( Edu+Fdv )^2+H^2dv^2=0 \)
or \( ( Edu+Fdv )^2=0 \) and \( H^2 dv^2=0 \)
or \( ( Edu+Fdv )^2=0 \) and \( dv^2=0 \)
or \(Edu+Fdv=0 \) and \( dv=0 \)
or \(Edu =0 \) and \( dv=0 \)
or \( du=0 \) and \( dv= 0 \) , which is not possible.
Hence, first fundamental form is positive definite in \( du \) and \( dv \).
Property 2 : First Fundamental Form is Invariant
The first fundamental form is invariant under parametric transformation.
Proof
Let \( S:\vec{r}=\vec{r}( u,v ) \) be a surface, and parameters \( ( u,v ) \) is transformed into another set \(( U,V ) \) set of parameters such that
\(U =\phi ( u,v ) \) and \( V=\psi ( u,v ) \)
Now
\(I = Edu^2+2Fdudv+Gdv^2 \) where \( E=\vec{r}_1^2,F=\vec{r}_1.\vec{r}_2,G=\vec{r}_2^2 \)
or
\( I =(\textcolor{red}{ \vec{r}_1} du+ \textcolor{red}{ \vec{r}_2} dv )^2 \)
or \( I =( \textcolor{red}{ \frac{\partial \vec{r}} {\partial u}} du + \textcolor{red}{ \frac{\partial \vec{r}} {\partial v}} dv)^2 \)
or \(I = ( [ \textcolor{red}{ \frac{\partial \vec{r}} {\partial U} \frac{\partial U} {\partial u} + \frac{\partial \vec{r}} {\partial V} \frac{\partial V} {\partial u} } ] du + [ \textcolor{red}{ \frac{\partial \vec{r}} {\partial U} \frac{\partial U} {\partial v} + \frac{\partial \vec{r}} {\partial V} \frac{\partial V} {\partial v} }] dv)^2 \)
or
\(I = ( \frac{\partial \vec{r}} {\partial U} \frac{\partial U} {\partial u} du + \frac{\partial \vec{r}} {\partial V} \frac{\partial V} {\partial u} du + \frac{\partial \vec{r}} {\partial U} \frac{\partial U} {\partial v} dv + \frac{\partial \vec{r}} {\partial V} \frac{\partial V} {\partial v} dv)^2 \)
or \(I = ( \frac{\partial \vec{r}} {\partial U} \frac{\partial U} {\partial u} du +\frac{\partial \vec{r}} {\partial U} \frac{\partial U} {\partial v} dv + \frac{\partial \vec{r}} {\partial V} \frac{\partial V} {\partial u} du + \frac{\partial \vec{r}} {\partial V} \frac{\partial V} {\partial v} dv)^2 \)
or \(I = ( \frac{\partial \vec{r}} {\partial U} [\frac{\partial U} {\partial u} du + \frac{\partial U} {\partial v} dv] + \frac{\partial \vec{r}} {\partial V} [ \frac{\partial V} {\partial u} du + \frac{\partial V} {\partial v} dv])^2 \)
or \(I = ( \frac{\partial \vec{r}} {\partial U}dU + \frac{\partial \vec{r}} {\partial V}dV)^2 \)
or \(I = EdU^2+2FdU dV+GdV^2 \) where \( E= (\frac{\partial \vec{r}}
{\partial U})^2,F=\frac{\partial \vec{r}} {\partial U}.\frac{\partial \vec{r}} {\partial V},G= (\frac{\partial \vec{r}} {\partial V})^2 \)
This shows that, first fundamental form is invariant under parametric transformation.
Elements of surface area
Let \(S:\vec{r}=\vec{r}( u,v )\) be a surface. If \(\Delta R=PQRS\) is a small region on the surface with
\(P( u,v ),Q( u+du,v ),R( u+du,v+dv )\) and \(( u,v+dv ) \).
Then, \( PQRS \) tends to a parallelogram when \( du, dv \) are very small and positive
Thus
Area of \( PQRS = \vec{PQ} \times \vec{PS}  \) (i)
Here
\( \vec{PQ}=\vec{OQ}\vec{OP} \)
or
\( \vec{PQ}=\vec{r}( u+du,v )\vec{r}( u,v ) \)
or
\( \vec{PQ}=\vec{r}( u,v )+\vec{r}_1du+... \vec{r}( u,v ) \)
Taking first order approximation, we get
\( \vec{PQ}=\vec{r}_1du \) (ii)
Similarly we get
\( \vec{PS}=\vec{r}_2dv \) (iii)
Thus, from (i), (ii) and (iii), we have
Area of \( PQRS =  \vec{r}_1 du \times \vec{r}_2dv  \)
or
Area of \( PQRS = \vec{r}_1 du \times \vec{r}_2 dv  \)
or
Area of \( PQRS = \vec{r}_1 \times \vec{r}_2  du dv\)
or
Area of \( PQRS =H du dv\)
Second Fundamental Form of the Surface
Let \(S:\vec{r}=\vec{r}( u,v )\) be a surface.Then quadratic differential form in \( du, dv \)\( II:Ldu^2+2Mdudv+Ndv^2 \) where \(L=\vec{r}_{11}.\vec{N},M=\vec{r}_{12}.\vec{N},N=\vec{r}_{22}.\vec{N} \)
is called second fundamental form of the surface.
The coefficients \(L,M,N \) are called the second fundamental coefficients or magnitude of second order.
Geometry of Second Fundamental Form of the Surface
Length of perpendicular on the tangent plane from neighborhood point on the surface is
\(\frac12( Ldu^2+2Mdudv+Ndv^2 ) \)
Proof
Let \( S:\vec{r}=\vec{r}( u,v ) \) be a surfacein which \( P \) and \( Q \) are two neighboring points with
\( \vec{OP}=\vec{r} \) and \( \vec{OQ}=\vec{r}+d\vec{r}\)
Then
\( \vec{PQ}=\vec{OQ}\vec{OP} \)
or \( d\vec{r}=\vec{r}( u+du,v+dv )\vec{r}( u,v) \)
or \( d\vec{r}=\vec{r}( u,v )+( \vec{r}_1du+\vec{r}_2dv )+\frac{1}{2!}( \vec{r}_1du+\vec{r}_2dv )^2+...\vec{r}( u,v ) \)
or \( d\vec{r}=( \vec{r}_1du+\vec{r}_2dv )+\frac{1}{2!}( \vec{r}_1du+\vec{r}_2dv )^2+... \)
The second order approximation is
\( d\vec{r}=( \vec{r}_1du+\vec{r}_2dv )+\frac{1}{2!}( \vec{r}_1du+\vec{r}_2dv )^2 \)
or \( d\vec{r}= (\vec{r}_1du+\vec{r}_2dv )+ \frac{1}{2} ( \vec{r}_{11} du^2 +2 \vec{r}_{12} du dv+ \vec{r}_{22} dv^2) \)
Now, let M be the projection of Q on the tangent plane at P, then
\( QM=\) Projection of \( \vec{PQ} \) on the normal at P
or \( QM=\vec{PQ}.\vec{N}\)
or \( QM=[ ( \vec{r}_1du+\vec{r}_2dv )+\frac{1}{2!}( \vec{r}_{11}du^2+2\vec{r}_{12}dudv+\vec{r}_{22}dv^2 ) ].\vec{N} \)
or \( QM=\frac{1}{2!}( \vec{r}_{11}.\vec{N}du^2+2\vec{r}_{12}.\vec{N}dudv+\vec{r}_{22}.\vec{N}dv^2 )\)
or \( QM=\frac{1}{2}( Ldu^2+2Mdudv+Ndv^2 )\) where \(L=\vec{r}_{11}.\vec{N},M=\vec{r}_{12}.\vec{N},G=\vec{r}_{22}.\vec{N} \)
Alternative forms of \( L, M, N \)
As we know that \(\vec{r_1}\) is tangent to ucurve; and \(\vec{r_2}\) is tangent to vcurve, so \(\vec{r_1}\) and \(\vec{r_2}\) are tangents of the surface \( S:\vec{r}=\vec{r}( u,v ) \). Therefore, \(\vec{r_1}\) and \(\vec{r_2}\) ate perpendicular to the normal vector \(\vec{N}\). Also, we know that \(L=\vec{r}_{11}.\vec{N},M=\vec{r}_{12}.\vec{N},G=\vec{r}_{22}.\vec{N} \). Now, the fundamental coefficients \(L,M,N\) can be alternatively explained as below.

Also, we know that
\(\vec{r}_1. \vec{N}=0 \)(1)
Differentiating (1) w. r. to u, we, get
\(\vec{r}_{11}. \vec{N}+\vec{r}_1. \vec{N}_1=0 \)
or \(L+\vec{r}_1. \vec{N}_1=0 \)
or \(\vec{r}_1. \vec{N}_1=L \) 
Also, we know that
\(\vec{r}_1. \vec{N}=0 \)(1)
Differentiating (1) w. r. to v, we, get
\(\vec{r}_{12}. \vec{N}+\vec{r}_1. \vec{N}_2=0 \)
or \(M+\vec{r}_1. \vec{N}_2=0 \)
or \(\vec{r}_1. \vec{N}_2=M \) 
Also, we know that
\(\vec{r}_2. \vec{N}=0 \)(1)
Differentiating (1) w. r. to u, we, get
\(\vec{r}_{21}. \vec{N}+\vec{r}_2. \vec{N}_1=0 \)
or \(M+\vec{r}_2. \vec{N}_1=0 \)
or \(\vec{r}_2. \vec{N}_1=M \) 
Also, we know that
\(\vec{r}_2. \vec{N}=0 \)(1)
Differentiating (1) w. r. to v, we, get
\(\vec{r}_{22}. \vec{N}+\vec{r}_2. \vec{N}_2=0 \)
or \(N+\vec{r}_2. \vec{N}_2=0 \)
or \(\vec{r}_2. \vec{N}_2=N \)
 \([\vec{r}_1,\vec{r}_2,\vec{r}_{11}]=\vec{r}_1 \times \vec{r}_2 .\vec{r}_{11}=H \vec{N} .\vec{r}_{11}=HL\)
 \([\vec{r}_1,\vec{r}_2,\vec{r}_{12}]=\vec{r}_1 \times \vec{r}_2 .\vec{r}_{12}=H \vec{N} .\vec{r}_{12}=HM\)
 \([\vec{r}_1,\vec{r}_2,\vec{r}_{21}]=\vec{r}_1 \times \vec{r}_2 .\vec{r}_{21}=H \vec{N} .\vec{r}_{21}=HM\)
 \([\vec{r}_1,\vec{r}_2,\vec{r}_{22}]=\vec{r}_1 \times \vec{r}_2 .\vec{r}_{22}=H \vec{N} .\vec{r}_{22}=HN\)
Nature of points on a surface
The second fundamental form of a surface measures osculation paraboloid, which helps to determines nature of points on the surface. These points are as follows Case 1: Parabolic
A point on a surface is called parabolic point if
\(LNM^2=0;L^2+M^2+N^2 \ne 0\)
In parabolic point, there exists a line in tangent plane whose normal curvature is zero. So, at this point, exactly one of principal curvatures \(K_1\) and \(K_2\) is zero. Since one normal curvature is zero, the direction corresponding to the zero principal curvature will be the direction of the asymptotic curve. The nbd point of the surface lies on same side of the tangent plane. An example of such point is shown in a paraboloid cylinder in figure.  Case 2: Hyperbolic
A point is called hyperbolic point if
\(LNM^2<0\)
In hyperbolic point, there exists two lines in tangent plane whose normal curvatures: \(K_1\) and \(K_2\) have opposite sign. In such point, there exists two asymptotic direction. The nbd point of the surface lies on both sides of the tangent plane. An example of such point is shown in a hyperboloid surface in figure  Case 3: Elliptic
A point is called elliptic point if
\(LNM^2>0\)
In elliptic point, there exists tangent plane whose normal curvatures: \(K_1\) and \(K_2\) have same sign. In such point, there exists no asymptotic direction. The nbd point of the lies on same side of the tangent plane. An example of such point is shown in an elliptic surface in figure  Case 4: Planner
A point is called planner point if
\(LNM^2=0;L^2+M^2+N^2= 0\)
If \(L,M,N\) and \(LNM^2\) are all zero, then surface is planar, means both of \(K_1\) and \(K_2\) is zero.
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