From 0f704e6dfacb902fd0c81ae96a1fff2ad58f5a59 Mon Sep 17 00:00:00 2001
From: bnewbold <bnewbold@robocracy.org>
Date: Sun, 24 Jan 2010 08:36:50 +0000
Subject: hmmmm, needs some work still

---
 math/tensors.page | 37 ++++++++++++++++++-------------------
 1 file changed, 18 insertions(+), 19 deletions(-)

(limited to 'math')

diff --git a/math/tensors.page b/math/tensors.page
index 7ea1848..feb9a01 100644
--- a/math/tensors.page
+++ b/math/tensors.page
@@ -1,33 +1,30 @@
 Tensors, Differential Geometry, Manifolds
 ============================================
 
-Most of this content is based on a 2002 Caltech course taught by Kip Thorn (PH237).
-
+*References: 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 ($P$), separation vectors ($\Delta \vector P$),
-curves ($Q(\zeta)$), tangent vectors ($\delta P / \delta \zeta \equiv
-\lim_{\Delta \zeta \rightarrow 0} \frac{ \vector{ Q(\zeta+\delta \zeta) -
-Q(\zeta) } }{\delta \zeta}$)
+curves ($Q(\zeta)$), tangent vectors 
+($\delta P / \delta \zeta \equiv \lim_{\Delta \zeta \rightarrow 0} \frac{  vector{ Q(\zeta+\delta \zeta) - Q(\zeta) } }{\delta \zeta}$)
 
-Coordinates: $\Chi^\alpha (P)$, where $\alpha = 0,1,2,3$;
-$Q(\Chi_0, \Chi_1, ...)$
-    there is an isomorphism between points and coordinates
+Coordinates: $\chi^\alpha (P)$, where $\alpha = 0,1,2,3$;
+$Q(\chi_0, \chi_1, ...)$
+there is an isomorphism between points and coordinates
 
-Coordinate basis: $\vector{e_\alpha} \equiv \left( \frac{\partial
-Q}{\partial \Chi^\alpha} \right$)
+Coordinate basis: 
+$$\vector{e_{\alpha}} \equiv \left( \frac{\partial Q}{\partial \chi^\alpha} \right$$
     
-    for instance, on a sphere with angles $\omega, \phi$: 
+for instance, on a sphere with angles $\omega, \phi$: 
     
-    $\vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta$
+$\vector{e_{\phi}} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_{\theta}$
 
 Components of a vector:
+$$\vector{A} = \frac{\partial P}{\partial \chi^\alpha }$$
 
-    $\vector{A} = \frac{\partial P}{\partial \Chi^\alpha }$
-
-Directional Derivatives: consider a scalar function defined on a manifold \Psi(P)
-    $\partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \Chi^\alpha}$
+Directional Derivatives: consider a scalar function defined on a manifold $\Psi(P)$:
+$$\partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \chi^\alpha}$$
 
 Mathematicians like to say that the coordinate bases are actually directional derivatives
 
@@ -46,7 +43,7 @@ inner product (aka dot product) to get a real.
 
 Define the **metric tensor** 
 $\bold{g}(\vector{A}, \vector{B}) = \vector{A} \cdot \vector{B}$. The 
-metric tensor is rank two and symetric (the vectors A and B could be swapped 
+metric tensor is rank two and symmetric (the vectors A and B could be swapped 
 without changing the scalar output value) and is the same as the inner product. 
 
 $$\Delta P \cdot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \cdot B = 1/4[ (A+B)^2 - (A-B)^2 ]$$
@@ -63,10 +60,12 @@ Spacetime
 Two types of vectors.
 
 Timelike: $\vector{\Delta P}$
-    $(\vector{\Delta P})^2 = -(\Delta \Tau)^2$
+:    $(\vector{\Delta P})^2 = -(\Delta \Tau)^2$
 
 Spacelike: $\vector{\Delta Q}$
-    $(\vector{\Delta Q})^2 = +(\Delta S)^2$
+:    $(\vector{\Delta Q})^2 = +(\Delta S)^2$
 
 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.
+
+[PH237]: http://elmer.tapir.caltech.edu/ph237/
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