User:T boyd/sandbox

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

$$ $$ \Delta S = (2x, 2y, 2z) = \vec N (x,y,z)

$$

defines a sphere of radius R in cartiesan coordinates.

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