path: root/math/tensors
diff options
Diffstat (limited to 'math/tensors')
1 files changed, 24 insertions, 16 deletions
diff --git a/math/tensors b/math/tensors
index 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}) $$`
@@ -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.