3 ms·
From the sounds of it, I think you'd be much more happy working with differential forms! Differential forms use only the wedge product and a duality mapping kno
by eigenspace 8y ago
From the sounds of it, I think you'd be much more happy working with differential forms! Differential forms use only the wedge product and a duality mapping known as the hodge dual (It's the GA equivalent of multiplying by the psuedoscalar). Differnetial forms easily supply 90% of the nice things in GA, so if you're unhappy with the geomtric product, I'd say that differnetial forms are perfect.
Geometric algebra will be here waiting for when you get tired of resorting to the hodge dual to take an inner product ;)
- ajkjk 8y agoI learned about differential forms in my physics degree, and then later discovered that the exterior algebra made a lot more sense if you introduced it separately. Basically it seems like the exterior algebra is useful in many settings, and differential-forms are just one rather confusing application of it. My current opinion is that exterior algebra should, someday, be taught first (around the time of linear algebra), and then later extended to differential forms when derivatives and manifolds get involved. I agree that the Hodge Dual is easily the worst part of exterior algebra. But you can treat the inner product as more fundamental, via *a ∧ b = <a,b> i. Either can essentially be constructed from the other (iirc).
- cygx 8y agoThe Hodge star isn't so bad: It's just the generalization of the idea that you can describe a plane by its normal vector.
- ajkjk 8y agoOn it's own it's not bad, but it's troublesome because given a complicated expression with wedges and stars, it's not at all intuitive how you can simplify it without working in coordinates.
- cygx 8y agoIs there a silver bullet, though? The complexity of having to mess around with epsilon tensors is still there no matter which abstraction you use to sweep it under the rug...