Don’t worry about the “mumbles” man, it’s just nerves. You have to remember that, when you’re recording, you’re not giving a huge life-determining speech in front of millions. Just envision that you’re explaining the concept to a close childhood friend. Slow down, calm your nerves, and gather your thoughts - small speech errors aren’t detracting from the information at all, trust me
0:06 You had me at geometric algebra ngl after watching video: I didn't really understand most of it (probably because I don't do quantum mechanics, but it's cool that you made this video as it seems highly motivating and encourages the use of geometric algebra due to very nice results even without STA, well done man
In my "SR in 3D Geometric Algebra" series, the last video solves two problems in Griffiths' Intro to Electrodynamics using the APS framework! There are unfortunately not many books or problems given specifically for the APS (it's so underappreciated imo), but you can totally use the APS to solve problems in non-APS textbooks. I do have a link to a paper with notes in the description, but I wrote that forever ago and haven't double-checked it for errors. You should check out the Bivector.net discord server, I'm active there and would be happy to help teach the APS when I'm able to.
Hi, great video! Though I did have a question about geometric (and generally Clifford) algebra that I can't seem to find answers to anywhere... a geometric algebra is a type of "algebra over a vector space" and so is a purely algebraic object. However, its operations are often associated with certain spatial transformations of manifolds, such as how sandwich product rotors are commonly used to represent rotations in Euclidean space. This extends to the operations of spacetime algebra being used to describe certain spatial transformations in the manifold of Minkowski space. But manifolds are not algebraic objects, so what gives us the mathematical "right", formally speaking, to imagine GA operations as manifold transformations, and even to imagine multivectors as being patches of directed area/volume/etc within the manifold?
I'm glad you enjoyed the video! Also, great question. Geometric Algebra can be thought of as purely algebraic, but thinking only in those terms can lead to confusion or distract from the importance of its geometric meaning. After all, it's called Algebra for a reason. As far as I'm aware (and at least in the algebra used in this video), the algebraic forms are not interpreted as spatial transformations of manifolds, but as geometric elements that exist within the space. I encourage you to use to term space over manifold here to avoid confusion. The convention is to use "space". A really good motivation for this is the Pythagorean Theorem. First, it is taken that a vector times itself returns its squared length: ^2=||^2=^2. Then if you add two orthogonal vectors such that +=, then (+)^2=^2+++^2=^2. But the Pythagorean Theorem says that ^2+^2=^2, which is only true if +=0. This means that there's a new geometric element created from two perpendicular vectors that is equal to the negative of its reverse product: =-. Well, in (general) space, a plane is constructed by two vectors, and reversing their order reverses their direction! Therefore the natural geometric interpretation of the product is an oriented plane segment. While this can be treated purely algebraically, it has inherent geometric meaning! Also, to speak on your point of manifolds not being algebraic objects, you don't even need to worry about that because the algebra isn't manipulating manifolds, but geometric, algebraic, entities within the space! So in essence, what gives it the "right" is the definition of the inner product and the assumption vectors represent geometric objects. There are abstract algebra ways to define it as well which some consider more "fundamental", but here they're not needed. Your idea of imagining patches of multivector elements within a manifold is on the right track. I'd word it more like: In a geometric algebra, a general multivector has discrete geometric elements within the space.
@@EccentricTuber Thanks for in depth response, it helped a lot! I only ask because after learning about Lie groups my whole perspective on how algebra and geometry can relate has changed. Unlike with geometric algebra, the way algebraic and geometric concepts intersect in a Lie group is part of its very definition. Makes me wonder if there's any connection or way to make a connection between GA and Lie groups, especially because spinors can be defined using either of them in certain ways
@@thevikifalcon7670oh there are definitely connections! For example, in this video the Lorentz transformations created by boosts and rotations form the Lorentz Group (which is a Lie group)! And the Dirac spinor shown towards the end transforms under this group.
@thevikifalcon7670 there are even algebraic objects that themselves 'shape out' manifolds through their trajectories, called lie groups and their associated lie algebras. (these sound a bit complex - no pun intended - but here's a really down-to-earth example: e^ix, i.e. the 'circle group' or U(1)) Manifolds and algebraic objects are not mutually exclusive things: a manifold may or may not have other structure on it. For instance, not all manifolds are even necessarily differentiable (all manifolds are 'topological manifolds', but not all of them have structure that supports doing calculus on them, c.f. the wiki link on 'topological manifolds')
@@EccentricTuber Lie groups are dual to Lie algebras (tangent plane). Space is dual to time -- Einstein. Space/time symmetries are dual to Mobius maps -- stereographic projection. The inner product is dual to the outer product (wedge product) synthesizes the geometric product -- Clifford Algebra. Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Injection is dual to surjection synthesizes bijection or isomorphism. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Categories (form, syntax) are dual to sets (substance, semantics). "Always two there are" -- Yoda.
I really dont understand how you map the unit scalar of a multivector to the time basis vector. I think it should make more sense to generate a clifford algebra for the minkowsky space if you would apply clifford algebra to special relativity.
The Clifford algebra generated by the Minkowski metric is known as the spacetime algebra, and the algebra of physical space described here is isomorphic (as an algebra) to the even sub-algebra of the spacetime algebra. Let e denote the basis blades of APS and g of STA. Then, we can explicitly model the isomorphism as the one induced by: e_x g_tx e_y g_ty e_z g_tz In essence, the isomorphism is given by including an extra factor of t in each basis. For time, this would be g_tt, which is just 1. That's why the scalar is mapped to the time basis vector
Are there any sources you can recommend to help me understand the special relativity and electromagnetism sections, I understand the first few equations but I got lost because I actually haven’t done this kind of math before
@@londisibanda1421 If you need to learn about Special Relativity on its own, I think EigenChris has a series on it. If you're talking more about the math used in Geometric Algebra, I have some background videos and a series on it (and Electromagnetism) where I pretty much work through the stuff in this video! Baylis' research papers are good for learning the Algebra of Physical Space. I'd also recommend Introductory Electrodynamics by Griffith and then Spacetime Algebra as a Powerful Tool for Electromagnetism (by Dorst I think).
Never heard it called APS before, or a pauli representation. "Representation" usually means matrices, which we don't need in the slightest in GA. The first thing I love about GA is that it renders matrices irrelevant.
Representations can also be used in a non-matrix context. And the APS is a common name in literature. Also, presenting matrices in the video is to show that those who use matrices don't have to. I have reasons for showing things in my video.
The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics. Lie groups are dual to Lie algebras (tangent plane). Space is dual to time -- Einstein. Space/time symmetries are dual to Mobius maps -- stereographic projection. The inner product is dual to the outer product (wedge product) synthesizes the geometric product -- Clifford Algebra. Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Injection is dual to surjection synthesizes bijection or isomorphism. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Categories (form, syntax) are dual to sets (substance, semantics). "Always two there are" -- Yoda.
Space/time symmetries (rotations, boosts) are dual to Mobius maps -- stereographic projection. Particles are dual to antiparticles, spin up is dual to spin down -- the Dirac equation. Space is dual to time -- Einstein. "Always two there are" -- Yoda. The magnetic field only exists for moving particles with a velocity!
Are you planning to go through the Dirac spinor in STA? I've been reading the paper on it and it needs a video and visuals. I'm thinking if I can't get anyone to do it, I might take a crack at it. However, I am not the ideal messenger as I don't have a PhD in anything, I'm not a mathematician or a physicist, I just use GA for graphics work and love physics. Thus, there's a lot of potential for me to make mistakes. Nevermind, looks like you have covered the paper. Still, I'd like to see more visual intuition.
@@EccentricTuber Space/time symmetries (rotations, boosts) are dual to Mobius maps -- stereographic projection. Particles are dual to antiparticles, spin up is dual to spin down -- the Dirac equation. Space is dual to time -- Einstein. "Always two there are" -- Yoda. The magnetic field only exists for moving particles with a velocity! Electro is dual to magnetic -- photons are dual.
Chirality exists in spacetime weyl/dirac but not in Pauli. I would like to hear how you deal with right-handed and left-handed ness in pauli to dirac. Sorry if this is a bit garbled. Love the video by the way
Chirality is dual to Helicity. Lie groups are dual to Lie algebras (tangent plane). Space is dual to time -- Einstein. Space/time symmetries are dual to Mobius maps -- stereographic projection. The inner product is dual to the outer product (wedge product) synthesizes the geometric product -- Clifford Algebra. Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Injection is dual to surjection synthesizes bijection or isomorphism. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Categories (form, syntax) are dual to sets (substance, semantics). "Always two there are" -- Yoda.
I'm somewhat self-conscious about whenever I misspeak words so it's mostly a mechanism I use to get myself to stop worrying about it. It takes me so many tries to get a recording where I'm not fumbling over every other word.
@@EccentricTuber If I did that, my videos would have nothing but mumble counters. I have the same trouble when I attempt to narrate a video that has no audio, and speak like I'm drunk or have dementia (but strangely, not if I'm making an ad-hoc recording using a tablet and not thinking about the words that I'm saying.)
@@EccentricTuber Especially if you're self-conscious about your voiceover, I'd recommend giving your video real subtitles. It's very easy, you just need to paste your script in and youtube will automatically sync it to the voice.
7:05 "The determinant of any matrix [...] is the matrix itself times its adjoint." Huh? That sounds very wrong. Such product isn't even a scalar (or anything that looks like a scalar).
Lie groups are dual to Lie algebras (tangent plane). Space is dual to time -- Einstein. Space/time symmetries are dual to Mobius maps -- stereographic projection. The inner product is dual to the outer product (wedge product) synthesizes the geometric product -- Clifford Algebra. Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Injection is dual to surjection synthesizes bijection or isomorphism. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Categories (form, syntax) are dual to sets (substance, semantics). "Always two there are" -- Yoda.
This is a good question! First, I must admit that I have never really used the exterior calculus. I'm familiar with it and I know some of its applications, but I've never had to use it because I knew the Geometric Algebra application instead. In its own way, I think that response kinda speaks for the potential advantage of this formalism. Geometric Algebra is "essentially" a unification of the exterior calculus with traditional vector mechanics.
@@TheRevAlokSingh The Hodge star operator contains the metric, and furthermore, differential forms may take values in Clifford algebra, as well as in any Lie algebra. This is the reason one can describe non-Abelian Yang-Mills theories in a very concise manner using differential forms.
@@Rmeggedon Well Hodge star operator is called dual operator in GA, that is, it has mostly the same function. GA actually does what Forms and Tensors do, as it is a superstructure for both of them. As Lie Groups are for Discrete Groups of Symmetries.
Lie groups are dual to Lie algebras (tangent plane). Space is dual to time -- Einstein. Space/time symmetries are dual to Mobius maps -- stereographic projection. The inner product is dual to the outer product (wedge product) synthesizes the geometric product -- Clifford Algebra. Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Injection is dual to surjection synthesizes bijection or isomorphism. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Categories (form, syntax) are dual to sets (substance, semantics). "Always two there are" -- Yoda.
Don’t worry about the “mumbles” man, it’s just nerves. You have to remember that, when you’re recording, you’re not giving a huge life-determining speech in front of millions. Just envision that you’re explaining the concept to a close childhood friend. Slow down, calm your nerves, and gather your thoughts - small speech errors aren’t detracting from the information at all, trust me
Focus and patience is really important when you are trying to express and show your ideas.
0:06 You had me at geometric algebra ngl
after watching video: I didn't really understand most of it (probably because I don't do quantum mechanics, but it's cool that you made this video as it seems highly motivating and encourages the use of geometric algebra due to very nice results even without STA, well done man
Thank you! I'm glad you liked it. You caught on to the thoughts I was trying to convey perfectly!
This was the just the video I needed. I knew geometric algebra, but your video will help me start research in higher physics.
Love this.
If you have any examples of worked practical problems (rather than formulations) that would be really appreciated.
Thanks so much!
In my "SR in 3D Geometric Algebra" series, the last video solves two problems in Griffiths' Intro to Electrodynamics using the APS framework! There are unfortunately not many books or problems given specifically for the APS (it's so underappreciated imo), but you can totally use the APS to solve problems in non-APS textbooks. I do have a link to a paper with notes in the description, but I wrote that forever ago and haven't double-checked it for errors. You should check out the Bivector.net discord server, I'm active there and would be happy to help teach the APS when I'm able to.
This is mind blowing 🤯
Hi, great video! Though I did have a question about geometric (and generally Clifford) algebra that I can't seem to find answers to anywhere... a geometric algebra is a type of "algebra over a vector space" and so is a purely algebraic object. However, its operations are often associated with certain spatial transformations of manifolds, such as how sandwich product rotors are commonly used to represent rotations in Euclidean space. This extends to the operations of spacetime algebra being used to describe certain spatial transformations in the manifold of Minkowski space. But manifolds are not algebraic objects, so what gives us the mathematical "right", formally speaking, to imagine GA operations as manifold transformations, and even to imagine multivectors as being patches of directed area/volume/etc within the manifold?
I'm glad you enjoyed the video! Also, great question. Geometric Algebra can be thought of as purely algebraic, but thinking only in those terms can lead to confusion or distract from the importance of its geometric meaning. After all, it's called Algebra for a reason. As far as I'm aware (and at least in the algebra used in this video), the algebraic forms are not interpreted as spatial transformations of manifolds, but as geometric elements that exist within the space. I encourage you to use to term space over manifold here to avoid confusion. The convention is to use "space". A really good motivation for this is the Pythagorean Theorem. First, it is taken that a vector times itself returns its squared length: ^2=||^2=^2. Then if you add two orthogonal vectors such that +=, then (+)^2=^2+++^2=^2. But the Pythagorean Theorem says that ^2+^2=^2, which is only true if +=0. This means that there's a new geometric element created from two perpendicular vectors that is equal to the negative of its reverse product: =-. Well, in (general) space, a plane is constructed by two vectors, and reversing their order reverses their direction! Therefore the natural geometric interpretation of the product is an oriented plane segment. While this can be treated purely algebraically, it has inherent geometric meaning! Also, to speak on your point of manifolds not being algebraic objects, you don't even need to worry about that because the algebra isn't manipulating manifolds, but geometric, algebraic, entities within the space! So in essence, what gives it the "right" is the definition of the inner product and the assumption vectors represent geometric objects. There are abstract algebra ways to define it as well which some consider more "fundamental", but here they're not needed. Your idea of imagining patches of multivector elements within a manifold is on the right track. I'd word it more like: In a geometric algebra, a general multivector has discrete geometric elements within the space.
@@EccentricTuber Thanks for in depth response, it helped a lot! I only ask because after learning about Lie groups my whole perspective on how algebra and geometry can relate has changed. Unlike with geometric algebra, the way algebraic and geometric concepts intersect in a Lie group is part of its very definition. Makes me wonder if there's any connection or way to make a connection between GA and Lie groups, especially because spinors can be defined using either of them in certain ways
@@thevikifalcon7670oh there are definitely connections! For example, in this video the Lorentz transformations created by boosts and rotations form the Lorentz Group (which is a Lie group)! And the Dirac spinor shown towards the end transforms under this group.
@thevikifalcon7670 there are even algebraic objects that themselves 'shape out' manifolds through their trajectories, called lie groups and their associated lie algebras.
(these sound a bit complex - no pun intended - but here's a really down-to-earth example: e^ix, i.e. the 'circle group' or U(1))
Manifolds and algebraic objects are not mutually exclusive things: a manifold may or may not have other structure on it. For instance, not all manifolds are even necessarily differentiable (all manifolds are 'topological manifolds', but not all of them have structure that supports doing calculus on them, c.f. the wiki link on 'topological manifolds')
@@EccentricTuber Lie groups are dual to Lie algebras (tangent plane).
Space is dual to time -- Einstein.
Space/time symmetries are dual to Mobius maps -- stereographic projection.
The inner product is dual to the outer product (wedge product) synthesizes the geometric product -- Clifford Algebra.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Injection is dual to surjection synthesizes bijection or isomorphism.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Categories (form, syntax) are dual to sets (substance, semantics).
"Always two there are" -- Yoda.
I really dont understand how you map the unit scalar of a multivector to the time basis vector. I think it should make more sense to generate a clifford algebra for the minkowsky space if you would apply clifford algebra to special relativity.
The Clifford algebra generated by the Minkowski metric is known as the spacetime algebra, and the algebra of physical space described here is isomorphic (as an algebra) to the even sub-algebra of the spacetime algebra. Let e denote the basis blades of APS and g of STA. Then, we can explicitly model the isomorphism as the one induced by:
e_x g_tx
e_y g_ty
e_z g_tz
In essence, the isomorphism is given by including an extra factor of t in each basis. For time, this would be g_tt, which is just 1. That's why the scalar is mapped to the time basis vector
this remembers me a lot of "Geometric algebra"
That makes sense, it is Geometric Algebra after all!
Are there any sources you can recommend to help me understand the special relativity and electromagnetism sections, I understand the first few equations but I got lost because I actually haven’t done this kind of math before
@@londisibanda1421 If you need to learn about Special Relativity on its own, I think EigenChris has a series on it. If you're talking more about the math used in Geometric Algebra, I have some background videos and a series on it (and Electromagnetism) where I pretty much work through the stuff in this video!
Baylis' research papers are good for learning the Algebra of Physical Space. I'd also recommend Introductory Electrodynamics by Griffith and then Spacetime Algebra as a Powerful Tool for Electromagnetism (by Dorst I think).
Never heard it called APS before, or a pauli representation. "Representation" usually means matrices, which we don't need in the slightest in GA. The first thing I love about GA is that it renders matrices irrelevant.
Representations can also be used in a non-matrix context. And the APS is a common name in literature. Also, presenting matrices in the video is to show that those who use matrices don't have to. I have reasons for showing things in my video.
The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics.
Lie groups are dual to Lie algebras (tangent plane).
Space is dual to time -- Einstein.
Space/time symmetries are dual to Mobius maps -- stereographic projection.
The inner product is dual to the outer product (wedge product) synthesizes the geometric product -- Clifford Algebra.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Injection is dual to surjection synthesizes bijection or isomorphism.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Categories (form, syntax) are dual to sets (substance, semantics).
"Always two there are" -- Yoda.
Space/time symmetries (rotations, boosts) are dual to Mobius maps -- stereographic projection.
Particles are dual to antiparticles, spin up is dual to spin down -- the Dirac equation.
Space is dual to time -- Einstein.
"Always two there are" -- Yoda.
The magnetic field only exists for moving particles with a velocity!
Are you planning to go through the Dirac spinor in STA? I've been reading the paper on it and it needs a video and visuals. I'm thinking if I can't get anyone to do it, I might take a crack at it. However, I am not the ideal messenger as I don't have a PhD in anything, I'm not a mathematician or a physicist, I just use GA for graphics work and love physics. Thus, there's a lot of potential for me to make mistakes.
Nevermind, looks like you have covered the paper. Still, I'd like to see more visual intuition.
Yeah, I am planning to do it. It may be a few months til I finish the video.
@@EccentricTuber Space/time symmetries (rotations, boosts) are dual to Mobius maps -- stereographic projection.
Particles are dual to antiparticles, spin up is dual to spin down -- the Dirac equation.
Space is dual to time -- Einstein.
"Always two there are" -- Yoda.
The magnetic field only exists for moving particles with a velocity!
Electro is dual to magnetic -- photons are dual.
Chirality exists in spacetime weyl/dirac but not in Pauli. I would like to hear how you deal with right-handed and left-handed ness in pauli to dirac. Sorry if this is a bit garbled. Love the video by the way
Chirality is dual to Helicity.
Lie groups are dual to Lie algebras (tangent plane).
Space is dual to time -- Einstein.
Space/time symmetries are dual to Mobius maps -- stereographic projection.
The inner product is dual to the outer product (wedge product) synthesizes the geometric product -- Clifford Algebra.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Injection is dual to surjection synthesizes bijection or isomorphism.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Categories (form, syntax) are dual to sets (substance, semantics).
"Always two there are" -- Yoda.
What is your "Mumble Counter" about? I don't hear the mumbles.
I'm somewhat self-conscious about whenever I misspeak words so it's mostly a mechanism I use to get myself to stop worrying about it. It takes me so many tries to get a recording where I'm not fumbling over every other word.
@@EccentricTuber If I did that, my videos would have nothing but mumble counters. I have the same trouble when I attempt to narrate a video that has no audio, and speak like I'm drunk or have dementia (but strangely, not if I'm making an ad-hoc recording using a tablet and not thinking about the words that I'm saying.)
@@PeeterJootI know exactly what you mean! Yeah, maybe I went a bit overboard with the counter...
@@EccentricTuber Especially if you're self-conscious about your voiceover, I'd recommend giving your video real subtitles. It's very easy, you just need to paste your script in and youtube will automatically sync it to the voice.
Dope video 🤪👍
Dope comment!
7:05 "The determinant of any matrix [...] is the matrix itself times its adjoint."
Huh? That sounds very wrong. Such product isn't even a scalar (or anything that looks like a scalar).
en.m.wikipedia.org/wiki/Adjugate_matrix
Also, of the matrix has complex elements, the determinant can be complex. As seen in this video.
Lie groups are dual to Lie algebras (tangent plane).
Space is dual to time -- Einstein.
Space/time symmetries are dual to Mobius maps -- stereographic projection.
The inner product is dual to the outer product (wedge product) synthesizes the geometric product -- Clifford Algebra.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Injection is dual to surjection synthesizes bijection or isomorphism.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Categories (form, syntax) are dual to sets (substance, semantics).
"Always two there are" -- Yoda.
What is the advantage of this formalism on exterior calculus?
This is a good question! First, I must admit that I have never really used the exterior calculus. I'm familiar with it and I know some of its applications, but I've never had to use it because I knew the Geometric Algebra application instead. In its own way, I think that response kinda speaks for the potential advantage of this formalism. Geometric Algebra is "essentially" a unification of the exterior calculus with traditional vector mechanics.
Exterior algebra doesn’t use a metric but we have one. That metric is what distinguishes Clifford algebra by universal property from exterior algebra.
@@TheRevAlokSingh The Hodge star operator contains the metric, and furthermore, differential forms may take values in Clifford algebra, as well as in any Lie algebra. This is the reason one can describe non-Abelian Yang-Mills theories in a very concise manner using differential forms.
@@Rmeggedon Well Hodge star operator is called dual operator in GA, that is, it has mostly the same function. GA actually does what Forms and Tensors do, as it is a superstructure for both of them. As Lie Groups are for Discrete Groups of Symmetries.
Lie groups are dual to Lie algebras (tangent plane).
Space is dual to time -- Einstein.
Space/time symmetries are dual to Mobius maps -- stereographic projection.
The inner product is dual to the outer product (wedge product) synthesizes the geometric product -- Clifford Algebra.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Injection is dual to surjection synthesizes bijection or isomorphism.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Categories (form, syntax) are dual to sets (substance, semantics).
"Always two there are" -- Yoda.
Ogni cosa? Cala che vendi
Eh bien ouais, c'est ce que nous appelons un clickbait
@@EccentricTuber si oh! 😃