Manuscript XII-C@1 · XII-C.tex · section · dead1004

All first and second ramp derivatives

C1: the density representative and actual cubic source › All first and second ramp derivatives · lines 669–717

XII-C@1#eq:Ccoefficients, XII-C@1#eq:M2

Define

C^{μν α β} (q) = - q (q^{μν} q^{α β} - q^{μ (α} q^{β) ν}) = C_{0} + κ C_{1} + κ^{2} C_{2} + O (κ^{3}) .

The three coefficients are explicitly

C_{0}^{μν α β} C_{1}^{μν α β} C_{2}^{μν α β} = η^{μν} η^{α β} - η^{μ (α} η^{β) ν}, = b_{1} C_{0}^{μν α β} + u_{1}^{μν} η^{α β} + η^{μν} u_{1}^{α β} - u_{1}^{μ (α} η^{β) ν} - η^{μ (α} u_{1}^{β) ν}, = b_{2} C_{0}^{μν α β} + b_{1} (u_{1}^{μν} η^{α β} + η^{μν} u_{1}^{α β} - u_{1}^{μ (α} η^{β) ν} - η^{μ (α} u_{1}^{β) ν}) + u_{2}^{μν} η^{α β} + η^{μν} u_{2}^{α β} + u_{1}^{μν} u_{1}^{α β} - u_{2}^{μ (α} η^{β) ν} - η^{μ (α} u_{2}^{β) ν} - u_{1}^{μ (α} u_{1}^{β) ν} .

Also define the quadratic tensor

M_{2}^{μν ρ} = C_{0}^{μν α β} Γ_{2}^{ρ}_{α β} + C_{1}^{μν α β} Γ_{1}^{ρ}_{α β} .

Displayed formulas in this passage

XII-C@1#eq:Ccoefficients · line 681
$C_{0}^{μν α β} C_{1}^{μν α β} C_{2}^{μν α β} = η^{μν} η^{α β} - η^{μ (α} η^{β) ν}, = b_{1} C_{0}^{μν α β} + u_{1}^{μν} η^{α β} + η^{μν} u_{1}^{α β} - u_{1}^{μ (α} η^{β) ν} - η^{μ (α} u_{1}^{β) ν}, = b_{2} C_{0}^{μν α β} + b_{1} (u_{1}^{μν} η^{α β} + η^{μν} u_{1}^{α β} - u_{1}^{μ (α} η^{β) ν} - η^{μ (α} u_{1}^{β) ν}) + u_{2}^{μν} η^{α β} + η^{μν} u_{2}^{α β} + u_{1}^{μν} u_{1}^{α β} - u_{2}^{μ (α} η^{β) ν} - η^{μ (α} u_{2}^{β) ν} - u_{1}^{μ (α} u_{1}^{β) ν} .$
XII-C@1#eq:M2 · line 710
$M_{2}^{μν ρ} = C_{0}^{μν α β} Γ_{2}^{ρ}_{α β} + C_{1}^{μν α β} Γ_{1}^{ρ}_{α β} .$
unlabeled · line 672
$C^{μν α β} (q) = - q (q^{μν} q^{α β} - q^{μ (α} q^{β) ν}) = C_{0} + κ C_{1} + κ^{2} C_{2} + O (κ^{3}) .$