Changes

=More About Coordinate Systems=

=More About Coordinate Systems=

The study of various coordinate systems (they are sometimes called "curvilinear coordinate systems") is central to the subject of multivariable calculus.  You are already familiar with some of the common coordinate systems such as polar coordinates in 2 dimensions and spherical or cylindrical coordinates in 3 dimensions.  In this section we will examine the general topic of coordinate systems, and how one can perform calculations—involving vectors, surfaces, integrals, and so on— directly in specialized coordinate systems.  Physical problems can often be solved more easily in a coordinate system that matches the symmetries of the problem.

The study of various coordinate systems (sometimes called "curvilinear coordinate systems") is central to the subject of multivariable calculus.  You are already familiar with some of the common coordinate systems such as polar coordinates in 2 dimensions, and spherical or cylindrical coordinates in 3 dimensions.  In this section we will examine the general topic of coordinate systems, and how one can perform calculations—involving vectors, surfaces, integrals, and so on—directly in specialized coordinate systems.  Physical problems can often be solved more easily in a coordinate system that matches the symmetries of the problem.

An extremely important consideration to be aware of is that geometrical or physical phenomena (points, vectors, regions, curves, surfaces, and integrals) have a fundamental existence and meaning that is independent of any coordinate system.  Coordinate systems simply give us ways to attach numbers to it.  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.

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.

More generally, it is:

More generally, it is:

::<math>\frac{\partial(f_1,f_2,...,f_n)}{\partial(x_1,x_2,...,x_n)} = \begin{bmatrix} \displaystyle\frac{\partial f_1}{\partial x_1} & ... & \displaystyle\frac{\partial f_1}{\partial x_n} \\ ... &  & ... \\ \displaystyle\frac{\partial f_3}{\partial x_1} &...&\displaystyle\frac{\partial f_n}{\partial x_n}\end{bmatrix} </math>

::<math>J(f_1,f_2,...,f_n / x_1,x_2,...,x_n) = \frac{\partial(f_1,f_2,...,f_n)}{\partial(x_1,x_2,...,x_n)} = \begin{bmatrix} \displaystyle\frac{\partial f_1}{\partial x_1} & ... & \displaystyle\frac{\partial f_1}{\partial x_n} \\ ... &  & ... \\ \displaystyle\frac{\partial f_3}{\partial x_1} &...&\displaystyle\frac{\partial f_n}{\partial x_n}\end{bmatrix} </math>

and the determinant is:

and the determinant is:

::<math>\begin{vmatrix} \frac{\partial f_1}{\partial x_1} & ... & \frac{\partial f_1}{\partial x_n} \\ ... &  & ... \\ \frac{\partial f_3}{\partial x_1} &...&\frac{\partial f_n}{\partial x_n}\end{vmatrix} </math>

::<math>\begin{vmatrix} \frac{\partial f_1}{\partial x_1} & ... & \frac{\partial f_1}{\partial x_n} \\ ... &  & ... \\ \frac{\partial f_3}{\partial x_1} &...&\frac{\partial f_n}{\partial x_n}\end{vmatrix} </math>

For polar coordinates, the Jacobian matrices are:

For polar coordinates, the Jacobian matrices are:

::<math>J(r, \theta / x, y) = \frac{\partial(r, \theta)}{\partial(x, y)} = \begin{bmatrix} \frac{\partial r}{\partial x} & \frac{\partial r}{\partial y} \\ \frac{\partial \theta}{\partial x} & \frac{\partial \theta}{\partial y} \end{bmatrix}</math>

::<math>J(r, \theta / x, y) = \frac{\partial(r, \theta)}{\partial(x, y)} = \begin{bmatrix} \displaystyle\frac{\partial r}{\partial x} & \displaystyle\frac{\partial r}{\partial y} \\ \\ \displaystyle\frac{\partial \theta}{\partial x} & \displaystyle\frac{\partial \theta}{\partial y} \end{bmatrix}</math>

and:

and:

::<math>J(x, y / r, \theta) = \frac{\partial(x, y)}{\partial(r, \theta)} = \begin{bmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \end{bmatrix}</math>

::<math>J(x, y / r, \theta) = \frac{\partial(x, y)}{\partial(r, \theta)} = \begin{bmatrix} \displaystyle\frac{\partial x}{\partial r} & \displaystyle\frac{\partial x}{\partial \theta} \\ \\ \displaystyle\frac{\partial y}{\partial r} & \displaystyle\frac{\partial y}{\partial \theta} \end{bmatrix}</math>

Working these out, we have

Working these out, we have

It follows from this that, if the second change is just the inverse of the first, bringing us back to the original coordinate system, the two matrices are inverses.

It follows from this that, if the second change is just the inverse of the first, bringing us back to the original coordinate system, the two matrices are inverses.

==Vectors in Arbitrary Coordinate Systems==

==Vectors in Arbitrary Coordinate Systems==