From f61026119df4700f69eb73e95620bc5928ca0fcb Mon Sep 17 00:00:00 2001 From: User Date: Tue, 13 Oct 2009 02:52:09 +0000 Subject: Grand rename for gitit transfer --- math/tensors | 74 ------------------------------------------------------------ 1 file changed, 74 deletions(-) delete mode 100644 math/tensors (limited to 'math/tensors') diff --git a/math/tensors b/math/tensors deleted file mode 100644 index d46810e..0000000 --- a/math/tensors +++ /dev/null @@ -1,74 +0,0 @@ -============================================ -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. -- cgit v1.2.3