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→More About Coordinate Systems: More emphasis on geometric universality. If you get the idea that I'm fanatical about this, you're right!
An extremely important consideration to be aware of is that geometrical or physical phenomena (points, vectors, regions, surfaces, etc.) have a fundamental existence and meaning that is independent of any coordinate system. Coordinate systems simply give us ways to attach numbers to them. Different coordinate systems will attach different numbers to a given vector field, for example, but the vector field has an underlying geometric meaning that does not change. The various vector field operations have an underlying geometrical or physical meaning, and coordinate systems simply let us perform mathematical calculations on them.
An extremely important consideration to be aware of is that geometrical or physical phenomena (points, vectors, regions, surfaces, etc.) have a fundamental existence and meaning that is independent of any coordinate system. Coordinate systems simply give us ways to attach numbers to them. Different coordinate systems will attach different numbers to a given vector field, for example, but the vector field has an underlying geometric meaning that does not change. The various vector field operations have an underlying geometrical or physical meaning, and coordinate systems simply let us perform mathematical calculations on them.
You have already seen the notion of "geometrical universality" in the definitions of the dot product and the cross product. They were given purely geometrical definitions that can be visualized without regard to any coordinate system. Than, for any Cartesian coordinate system, these operations were defined in terms of the components of the vectors in that system, and it was shown that those results matched the purely geometrical definitions. What we are going to do next is develop the tools to define vectors, and the dot product and cross product, in ''any coordinate system at all''. Later, we will introduce new operations—the divergence, the curl, and various integrals, and show how to calculate them in any coordinate system.
For example, Stokes' theorem makes a very powerful geometric statement about integrals of various vector fields over very general surfaces and lines. The concepts appearing in the theorem are purely geometric, and therefore independent of the coordinate system. Once we know how to manipulate those concepts (vector fields, integrals, and the "curl" operator) in any coordinate system, we will be able to choose a coordinate system that makes the surface simple. Specifically, we will change from the usual x/y/z Cartesian system to a u/v/w system for which the surface is defined by holding w=0. (Another way of looking at this is that the surface has been "parameterized" in terms of parameters u and v.) Once this is done, we will be able to calculate the curl operator, perform the integration, and prove Stokes' theorem in the u/v/w system.
For example, Stokes' theorem makes a very powerful geometric statement about integrals of various vector fields over very general surfaces and lines. The concepts appearing in the theorem are purely geometric, and therefore independent of the coordinate system. Once we know how to manipulate those concepts (vector fields, integrals, and the "curl" operator) in any coordinate system, we will be able to choose a coordinate system that makes the surface simple. Specifically, we will change from the usual x/y/z Cartesian system to a u/v/w system for which the surface is defined by holding w=0. (Another way of looking at this is that the surface has been "parameterized" in terms of parameters u and v.) Once this is done, we will be able to calculate the curl operator, perform the integration, and prove Stokes' theorem in the u/v/w system.
::::***** Need to put in some examples, etc. I have them worked out, but typing them in is tedious.
::::***** Need to put in some examples, etc. I have them worked out, but typing them in is tedious.
=Problems=
=Problems=