User:T boyd/sandbox

A surface can be defined implicitly as ${\displaystyle S(c1,c2,c3)=s}$. For example: ${\displaystyle S(x,y,z)=x^{2}+y^{2}+z^{2}=R^{2}}$ defines a sphere of radius R in cartiesan coordinates. ${\displaystyle S(r,\theta ,\phi )=r=R^{2}}$ defines a sphere of radius R in spherical polar coordinates.

The area element of a coordinate system is the cross product of two orthogonal vectors parallel to the surface at a particular point.

It is necessary to find the normal vector for the surface, from which an arbitrary surface vector can be derived by the [Gram-Schmidt process].

From one surface vector, a second surface vector can be derived.

eg. For: ${\displaystyle S(x,y,z)=x^{2}+y^{2}+z^{2}=R^{2}dS=2x\cdot dx+2y\cdot dy+2z\cdot dz=0;{\frac {dy}{dx}}={\frac {2x}{y}}+{\frac {2z}{y}}\cdot dz=0;{\frac {dz}{dx}}={\frac {2x}{z}}+{\frac {2z}{y}}\cdot dy=0;{\vec {S}}_{1}=(dx,dy/dx,dz/dx);{\vec {S}}_{2}=(dx/dy,dy/dx,dz/dx);}$ ${\displaystyle \delta S=(2x,2y,2z)={\vec {N}}(x,y,z)}$

defines a sphere of radius R in cartiesan coordinates.

${\displaystyle S(r,\theta ,\phi )=r=R^{2}}$ defines a sphere of radius R in spherical polar coordinates.