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: Failed to parse (unknown function "\n"): {\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\n \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.