Mordred

Useful +1 was wondering a few times how to do chem latex

I concur +1

Lol there's some debate on whether math is a science or not.

It could also be argued there is no hard and fast truth in science. There is truth to the best of current understanding. Good example that everyone is familiar with in physics is Newtons laws of inertia. Everyone firmly believed the equations applied regardless of the measured objects inertia. Later findings showed its only valid for non relativistic inertia hence GR. I also wonder why this thread is in politics.

There's another key detail when it comes to the intrinsic curvature it is independent of any higher dimensional embedding. Very useful for invariant functions. Particularly when it comes to applying the tangent vector to the line element ds^s. Of key note is the basis vectors. Taking the infinisimal distance between P and Q (local) this can be shown independent on coordinate transformations. So the basis vectors are independent. Subsequently this equates to the covariant and contravarient vectors. As well as the Christoffel connections.

Its the set that can be continuously parameterized where each parameter is a coordinate. Line segment is one example. The association of points/coordinates with their measured values can be thought as the mappings of the manifold. However you may not be able to parameretize the entire manifold with the same parameters. Some manifolds are degenerate. Simple case a finite set of R^n in Euclidean space is non degenerate. However in Cartesian coordinates involving angle the origin or center is degenerate as at zero the angle is indeterminate. This is where the use of coordinate patches get involved. A manifold can have different coordinate systems as per above on the same manifold. With no preference to any coordinate system. The set of coordinate patches that covers the entire manifold is called an atlas. The saddle shape for negative curvature would be a good example. Edit scratch that last example it can be continously parameterised under the same coordinate set. The Cartesian coordinate requires 2 sets.For reasons provided above. Hyperbolic paraboloid \[z=x^2-y^2\] can be parameterized by one coordinate set. Though multiple sets can optionally be used it isn't required.

In the first example when you set the lines on a graph paper prior to bending this is intrinsically flat ( it is independant ) Once you curl the paper your curve is extrinsic as you need an extra dimension in order to curl the plane. Im not sure you missed anything tbh. Cylinder can simply be described as Eucludean flat is the internal geometry with extrinsic curvature. A sphere for example however has an intrinsic positive gaussian curvature ie circumference of the sphere. Intrinsic curvature K=1/r^2. With extrinsic curvature you need a higher dimension embedding. the 2 principle curvatures being \(k_1=K_2=1/R\) with mean curvature being \(H=1/2 (k_1+k_2)=1/R\). with \(K_{a,b}\) being the second fundamental form \[K_{\theta\theta}=R\] \[K_{\phi\phi}=Rsin^2\Theta\] \[k_{\theta\phi}=0\] under GR the extrinsic curvature tensor is the projection of the gradient of the hypersurface. \[K_{a,b}=-\nabla_\mu^\nu\] \[K_{\theta\theta}=\frac{r}{\sqrt{g(r)}}\] \[K_{\phi\phi}=\frac{r\sin^2\theta}{\sqrt{g(r)}}\] mean curvature bieng \[k=h^{a,b}k_{a,b}=\frac{2}{r\sqrt{g(r)}}\] K being a surface of a hypersphere where all affine normals intersect at the center above ties into n sphere aka hypersphere https://en.wikipedia.org/wiki/3-sphere edit: I was at work earlier decided when I got home to go into greater detail further detail in same format as above https://en.wikipedia.org/wiki/Gaussian_curvature https://faculty.sites.iastate.edu/jia/files/inline-files/gaussian-curvature.pdf https://arxiv.org/pdf/1209.3845

Nice explanation @KJW +1

Nice thread I may look into including Fock and Hilbert spaces into this thread might be handy to have specific spaces inclusive.

Good point ( pun intended)

Im going to a phenomena in SR in terms of symmetry that will likely blow your mind and quite frankly that of many other forum members. If one takes a manifold and In general, manifold is any set that can be continuously parameterised. The number of independent parameters required to specify any point in the set uniquely is the dimension of the manifold, and the parameters themselves are the coordinates of the manifold. so lets use a simple manifold a sheet of graph paper. Now on that graph paper place two points label one point P and the other Q. The actual label doesn't matter. Now measure the distance between P and Q. Then roll up the paper into a cylinder. Now this is the tricky part. Following the surface of the paper the distance does not change between P and Q. The geometry is still Euclidean but now in cylindrical coordinates. So now we introduce two terms " intrinsic geometry and extrinsic geometry. The above case the ds^2 (seperation distance ) is unchanged so it is invariant. The geometry intrinsically is identical to Euclidean flat. Aka the laws of physics is the same regardless of inertial reference frame. There is no intrinsic curvature in this case. The curvature itself is the extrinsic geometry ( the cylinder viewed from the outside) in the first case think of an ant embedded on its surface. The above is essential to understand symmetry relations in SR and GR unfortunately when you combine time dilation using the Interval (ct) and apply the Lorentz transformations the example above becomes more complex as the above is a 3 dimensional manifold while spacetime is described by 4 dimensional manifolds. However there is no limit to the number of ubique parameters that can be used as unique coordinates on a manifold. Aka higher dimensions. ( a parameter can be any set) ie set representing time or charge or temperature etc etc. The above is something you need the math skills to properly understand and the above is also needed to understand SR ( Minkowskii metric) GR via the field equations including its tensors and gauge gauge groups. A common term for the above is local vs global geometry. For others the above is an example of coordinate basis. However the parameters used and subsequent coordinates can be under others "basis". The above should also give a very strong hint of why covariant and contravariant vectors become useful on manifolds 4d and up. (Kronecker delta ) first case 4d needing ( Levi-Cevita ) the above is also useful with regards to Hilbert spaces aka QM. The above is obviously a 2d manifold mathematical the extrinsic dimensions however requires the z axis to (curl) the 2d plane. ( curl equates to rotational symmetry)

Lets try a simple example with regards to symmetry relations involved for spacetime. The Minkowskii metric for example has a specific mathematical statement defining orthogonality which must also be symmetric. \[\mu \cdot \nu= \nu \cdot \mu\] This directly applies to vectors more specifically covariant and contravariant vectors. You wont find any image that will teach the above key relationship. In your first link where is the length contraction as applied under the Lorentz transformations? To give another example

I wouldn't rely on images to understand gravity. They can often be more misleading. For example describe how either image shows the equivalence principle between inertia mass and gravitational mass or show how either image describes time dilation when neither image contains a spacetime diagram Lets put it this way and only you can honestly answer the following. In the time frame since your last post in the your other thread has your understanding of gravity significantly improved ? In that same time frame would your understanding of gravity improved significantly more if you had instead studied an introductory GR textbook such as Lewis Ryder's General relativity even if you only spend 2 to 3 hours on it a week ?

In order to understand gravity especially under GR you need to have a good grasp of kinematics. GR uses the 4 momentum and its symmetry relations are freefall states with no force acting upon the object or particle ( which directly applies to the conservation of momentum). Newton treats gravity as a force acting upon the falling object instead of freefall. GR uses spacetime curvature instead of treating gravity as a force. Curvature is easily understood if you take 2 or more freefall paths. For example take 2 laser beams in parallel. If the two beams remain parallel spacetime is flat. If the two beams converge it is positively curved. If the beams diverge ( move apart) = negative curvature. To better understand freefall study the Principle of equivalence. https://webs.um.es/bussons/EP_lecture.pdf You can see under the section " Local inertial frames" the freefall paths are approaching one another as the elevator is freefalling toward a center of mass ( positive curvature). Indeed the equation of GR employ geodesics to describe these paths for photons they are null geodesics and how parallel null geodesics remain parallel converge of diverge are used to describe the curvature terms. At a more advanced level this is the basis of the Raychaudhuri equations. Which is a good formalism to understand how spacetime geometry affects multiparticle paths with regards to curvature terms. As mentioned Newton described gravity as a force so the falling objects have the gravitional force acting upon them. Under GR they are in freefall but the spacetime paths become curved. Hence gravity is treated as the result of spacetime curvature . In terms of geometry the Newton case the geometry is Euclidean and unchanging. This isn't the case under GR. In GR the geometry itself changes resulting in what we describe as gravity.

I agree pipelines to both coasts only makes economic sense provided the relevant safeties vs leakage and detection are properly installed. Lol though in our times detection of theft would also be required Nice descriptive lol