Major deficiency in Physics[Vectors]; Distinct sets of basis vectors are not recognized!

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Major deficiency in Physics[Vectors]; Distinct sets of basis vectors are not recognized!

Posted: nmacsai 260 Product: Maple 2024

December 12 2024

Major deficiency in Physics[Vectors]; Distinct sets of basis vectors are not recognized!

You can't define vectors in alternative bases like: {\hat{i}',\hat{j}',\hat{k}'} or {\hat{i}_{1},\hat{j}_{2},\hat{k}_{3}}.

This deficiency has been around for a while. I have found other posts regarding this problem.

The deficiency greatly reduces the allowable calculations with Physics[Vector].

Are there any plans to fix this?

Here is my example which shows this deficiency in more detail.

physics_vectors_and_multiple_unit_vectors.mw

[`&x`, `+`, `.`, Assume, ChangeBasis, ChangeCoordinates, CompactDisplay, Component, Curl, DirectionalDiff, Divergence, Gradient, Identify, Laplacian, Nabla, Norm, ParametrizeCurve, ParametrizeSurface, ParametrizeVolume, Setup, Simplify, `^`, diff, int]

(1)

Crucial Deficiency in Physics[Vectors]

I can only guess the purpose of the Physics[Vectors] package from reviewing it's corresponding help documentation. My interpretation of the documentation leads me to believe that the package is best used for generating vector equation formulas in different coordinate bases of a SINGLE coordinate system.

This means one can easily generate position vector expressions such as:

(1.1)

Cylindrical Position Vector

The position vector in a cylindrical basis is given by:

(1.1.1)

(1.1.2)

Spherical Position Vector

(1.2.1)

(1.2.2)

As is known from the vector analysis of curvilinear coordinate systems the basis vectors can depend on the coordinates in question.

In cylindrical, the basis vectors are

(1.2)

(1.3)

and in spherical, the basis vectors are

(1.4)

(1.5)

(1.6)

Example of this Deficiency using Biot-Savart Law

Biot-Savart law can be used to calculate a magnetic field due to a current carrying wire. The deficiency in question can be observed by explicity constructing the integrand in the Biot-Savart integral defined below.

In electrodynamics, quantum mechanics and applied mathematics, it is common practice to define a position of observation by a vector and a position of the source responsible for generating the field by a vector .

It is just as common to define the difference in these vectors as

(1.3.1)

and thus

(1.3.2)

as found in the integrand of the Biot-Savart integral.

It suffices to consider in a cylindrical basis for this argument.

The observation position is:

The source position is:

The deficiency in question arises because MAPLE cannot define multiple unit vectors in distinct bases such as or . These pairs of unit vectors arise naturally, as shown above in Biot-Savart law.

If we look at and generally, they look like:

If the bases vectors and are Cartesian and are not related related through rotations so that

and so,

the radial unit vectors in cylindrical are then,

In typical problems, the anglular location of the observation point, φ, is distinct from the angular location of the source, and so under this condition, .

Consider the classic problem of the magnetic field due to a circular current carrying wire. Surely, the angular coordinate of one location of the current carrying wire is different from the angular coordinate of an observation point hovering above and off-axis on the other side of the current carrying wire. See figure below.