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 diff --git a/math/tensors b/math/tensorsindex 42fa841..e15270a 100644--- a/math/tensors+++ b/math/tensors@@ -8,20 +8,28 @@ Tensors, Differential Geometry, Manifolds On a manifold, only "short" vectors exist. Longer vectors are in a space tangent to the manifold. -There are points (P), separation vectors (\Delta \vector P), curves ( Q(\zeta) ), tangent vectors ( \delta P / \delta \zeta \equiv \lim_{\Delta \zeta \rightarrow 0} \frac{ \vector{ Q(\zeta+\delta \zeta) - Q(\zeta) } }{\delta \zeta} )+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: \Chi^\alpha (P), where \alpha = 0,1,2,3; Q(\Chi_0, \Chi_1, ...)+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: \vector{e_\alpha} \equiv \left( \frac{\partial Q}{\partial \Chi^\alpha} \right)- for instance, on a sphere with angles \omega, \phi: - \vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta+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:- \vector{A} = \frac{\partial P}{\partial \Chi^\alpha }++ :m:$\vector{A} = \frac{\partial P}{\partial \Chi^\alpha }$ Directional Derivatives: consider a scalar function defined on a manifold \Psi(P)- \partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \Chi^\alpha}+ :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 @@ -32,24 +40,24 @@ A **tensor** :m:$\bold{T}$ has a number of slots (called it's **rank**), takes as an example for a rank-3 tensor: :m:$$\bold{T} ( \alpha \vector{A} + \beta \vector{B}, \vector{C}, \vector{D}) =- \alpha \bold{T} (\vector{A}, \vector{C}, \vector{D}) +- \beta \bold{T} (\vector{B}, \vector{C}, \vector{D})$$+\alpha \bold{T} (\vector{A}, \vector{C}, \vector{D}) + \beta \bold{T}+(\vector{B}, \vector{C}, \vector{D}) $$ Even a regular vector is a tensor: pass it a second vector and take the inner product (aka dot product) to get a real. -Define the **metric tensor** -:m:\bold{g}(\vector{A}, \vector{B}) = \vector{A} \dot \vector{B}. The +Define the **metric tensor ** +:m:\bold{g}(\vector{A}, \vector{B}) = \vector{A} \cdot \vector{B}. The metric tensor is rank two and symetric (the vectors A and B could be swapped without changing the scalar output value) and is the same as the inner product. -:m:$$\Delta P \dot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \dot B = 1/4[ (A+B)^2 - (A-B)^2 ]$$+:m:$$\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} \dot \vector{E})(\vector{B} \dot \vector{F})(\vecotr{C} \dot \vector{G}) $$ +:m:$$\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 --------------@@ -57,10 +65,10 @@ Spacetime Two types of vectors. Timelike: :m:$\vector{\Delta P}$- (\vector{\Delta P})^2 = -(\Delta \Tau)^2+ :m:$(\vector{\Delta P})^2 = -(\Delta \Tau)^2$ -Spacelike: \vector{\Delta Q}- (\vector{\Delta Q})^2 = +(\Delta S)^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.