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 diff --git a/math/tensors.page b/math/tensors.pagenew file mode 100644index 0000000..d46810e--- /dev/null+++ b/math/tensors.page@@ -0,0 +1,74 @@+============================================+Tensors, Differential Geometry, Manifolds+============================================++.. note:: Most of this content is based on a 2002 Caltech course taught by+ Kip Thorn [PH237]_+++On a manifold, only "short" vectors exist. Longer vectors are in a space tangent to the manifold.++There are points (:m:$P$), separation vectors (:m:$\Delta \vector P$),+curves (:m:$Q(\zeta)$), tangent vectors (:m:$\delta P / \delta \zeta \equiv+\lim_{\Delta \zeta \rightarrow 0} \frac{ \vector{ Q(\zeta+\delta \zeta) -+Q(\zeta) } }{\delta \zeta}$)++Coordinates: :m:$\Chi^\alpha (P)$, where :m:$\alpha = 0,1,2,3$;+:m:$Q(\Chi_0, \Chi_1, ...)$+ there is an isomorphism between points and coordinates++Coordinate basis: :m:$\vector{e_\alpha} \equiv \left( \frac{\partial+Q}{\partial \Chi^\alpha} \right$)+ + for instance, on a sphere with angles :m:$\omega, \phi$: + + :m:$\vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta$++Components of a vector:++ :m:$\vector{A} = \frac{\partial P}{\partial \Chi^\alpha }$++Directional Derivatives: consider a scalar function defined on a manifold \Psi(P)+ :m:$\partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \Chi^\alpha}$++Mathematicians like to say that the coordinate bases are actually directional derivatives++Tensors+------------++A **tensor** :m:$\bold{T}$ has a number of slots (called it's **rank**), takes a vector in each slot, and returns a real number. It is linear in vectors; +as an example for a rank-3 tensor:++:m:$$\bold{T} ( \alpha \vector{A} + \beta \vector{B}, \vector{C}, \vector{D}) =+\alpha \bold{T} (\vector{A}, \vector{C}, \vector{D}) + \beta \bold{T}+(\vector{B}, \vector{C}, \vector{D})$$++Even a regular vector is a tensor: pass it a second vector and take the +inner product (aka dot product) to get a real.++Define the **metric tensor** +:m:$\bold{g}(\vector{A}, \vector{B}) = \vector{A} \cdot \vector{B}$. The +metric tensor is rank two and symetric (the vectors A and B could be swapped +without changing the scalar output value) and is the same as the inner product. ++:m:$$\Delta P \cdot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \cdot B = 1/4[ (A+B)^2 - (A-B)^2 ]$$++Starting with individual vectors, we can construct tensors by taking the +product of their inner products with empty slots; for example++:m:$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\_ ,\_ ,\_)$$ +:m:$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \cdot \vector{E})(\vector{B} \cdot \vector{F})(\vecotr{C} \cdot \vector{G})$$ ++Spacetime+--------------++Two types of vectors.++Timelike: :m:$\vector{\Delta P}$+ :m:$(\vector{\Delta P})^2 = -(\Delta \Tau)^2$++Spacelike: :m:$\vector{\Delta Q}$+ :m:$(\vector{\Delta Q})^2 = +(\Delta S)^2$++Because product of "up" and "down" basis vectors must be a positive Kronecker +delta, and timelikes squared come out negative, the time "up" basis must be negative of the time "down" basis vector.