I’m telling you that after a few years working with the geometric product a whole bunch, the wedge product really doesn’t cut it.
> I still have basically no idea what 'AB' means when both are arbitrary-grade multivectors
Any mathematical language (or any natural language) can express a bunch of nonsensical and useless things. What is the sine of the square root of the logarithm of the tangent of some polynomial applied to a scalar? Who knows?!
You can certainly make up nonsensical expressions using matrices, differential forms, etc.
In a concrete problem, sometimes you have to think a bit to figure out what the geometrical meaning is of a particular multivector, but if the problem came out of a physical situation, I haven’t yet found a case where I couldn’t explain it.
Your multivector might be an electromagnetic field. It might be a rotor. It might be a representation of a sphere in the conformal model, ...
> I still have basically no idea what 'AB' means when both are arbitrary-grade multivectors
Any mathematical language (or any natural language) can express a bunch of nonsensical and useless things. What is the sine of the square root of the logarithm of the tangent of some polynomial applied to a scalar? Who knows?!
You can certainly make up nonsensical expressions using matrices, differential forms, etc.
In a concrete problem, sometimes you have to think a bit to figure out what the geometrical meaning is of a particular multivector, but if the problem came out of a physical situation, I haven’t yet found a case where I couldn’t explain it.
Your multivector might be an electromagnetic field. It might be a rotor. It might be a representation of a sphere in the conformal model, ...