The quick answer is that del M is just a piece of notation for the boundary of M - sometimes the same symbol gets used for different things. The long answer is that the operations of differentiation and of taking the boundary have a lot in common. For example, the boundary of A times B is (boundary of A) times B union A times (boundary of B), which looks exactly like Leibniz's rule for differentiating a product of you replace "boundary" by differentiation and "union" by addition. This is exploited in homology/cohomology theory: differentiation of differential forms allows you to define the de Rham complex for cohomology; boundary allows you to define the chain complex for ordinary homology: they're somehow dual operations. Compare with Stokes's theorem: it says that if f is a differential form and C is a chain then integrating df over C is the same as integrating f over the boundary of C. So the choice of similar notation is by design: to highlight this duality.
Photons, light, null rays or the electro-magnetic field = fixed points! Y = X. ct = x where c = 1 implies t = x. Y is equal to X, Y is the same, similar, equivalent or dual to X. Y is dual to X! Photons or light are dual, electro is dual to magnetic! All mathematical equations are dual! f(x) = ct = x, Brouwer's fixed point theorem proves that null rays or light are dual! Duality creates reality! The velocity of light is the same and equal for all observers hence fixed points conform to a principle of objective democracy. "Always two there are" -- Yoda.
@@iancheung3587 Points are dual to lines -- the principle of duality in geometry. Null lines (vectors, space/time) are dual to null points (spinors, twistor space) -- Roger Penrose. Space is dual to time -- Einstein. Twistor space is an equivalent or dual description of space/time (duality). Null lines or light rays are perpendicular or dual to themselves from our perspective -- self dual. The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Light rays or photons are pure energy so energy is duality, duality is energy! Potential energy is dual to kinetic energy -- gravitational energy is dual. Yin is dual to Yang. The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero -- non null homotopic. The big bang is an infinite negative curvature singularity (a point). Gaussian negative curvature is defined by two dual points:- en.wikipedia.org/wiki/Gaussian_curvature Dark energy is repulsive gravity (inflation) or negative curvature, hyperbolic space. Duality creates reality!
@@iancheung3587 The laws of physics conform to a principle of objective or absolute democracy as they are the same and equal for all observers. Objective is dual to subjective, absolute is dual to relative. Democracy is therefore dual. Agreement is dual to disagreement.
thanks, I'm trying to learn algebraic topology by myself and your content is amazing ty.
The volume is too soft. Hard to hear you
At first I wondered what "beactsmap" meant, but then I realized you were skipping spaces for some reason. :)
17:44 missed an a in the denominator, no big deal though any one watching knows what you meant. Beautiful proof presentation btw.
Doh! Thanks.
Thank you, thank you, thank you!!!!
How boundary is equal to del sir
The quick answer is that del M is just a piece of notation for the boundary of M - sometimes the same symbol gets used for different things. The long answer is that the operations of differentiation and of taking the boundary have a lot in common. For example, the boundary of A times B is (boundary of A) times B union A times (boundary of B), which looks exactly like Leibniz's rule for differentiating a product of you replace "boundary" by differentiation and "union" by addition. This is exploited in homology/cohomology theory: differentiation of differential forms allows you to define the de Rham complex for cohomology; boundary allows you to define the chain complex for ordinary homology: they're somehow dual operations. Compare with Stokes's theorem: it says that if f is a differential form and C is a chain then integrating df over C is the same as integrating f over the boundary of C. So the choice of similar notation is by design: to highlight this duality.
T99
Photons, light, null rays or the electro-magnetic field = fixed points!
Y = X.
ct = x where c = 1 implies t = x.
Y is equal to X, Y is the same, similar, equivalent or dual to X.
Y is dual to X!
Photons or light are dual, electro is dual to magnetic!
All mathematical equations are dual!
f(x) = ct = x, Brouwer's fixed point theorem proves that null rays or light are dual!
Duality creates reality!
The velocity of light is the same and equal for all observers hence fixed points conform to a principle of objective democracy.
"Always two there are" -- Yoda.
@Hyperduality lol what are u on XD
@@iancheung3587 Points are dual to lines -- the principle of duality in geometry.
Null lines (vectors, space/time) are dual to null points (spinors, twistor space) -- Roger Penrose.
Space is dual to time -- Einstein.
Twistor space is an equivalent or dual description of space/time (duality).
Null lines or light rays are perpendicular or dual to themselves from our perspective -- self dual.
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Light rays or photons are pure energy so energy is duality, duality is energy!
Potential energy is dual to kinetic energy -- gravitational energy is dual.
Yin is dual to Yang.
The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero -- non null homotopic.
The big bang is an infinite negative curvature singularity (a point).
Gaussian negative curvature is defined by two dual points:-
en.wikipedia.org/wiki/Gaussian_curvature
Dark energy is repulsive gravity (inflation) or negative curvature, hyperbolic space.
Duality creates reality!
@@iancheung3587 The laws of physics conform to a principle of objective or absolute democracy as they are the same and equal for all observers.
Objective is dual to subjective, absolute is dual to relative.
Democracy is therefore dual.
Agreement is dual to disagreement.
39. 1 T8899