228x Filetype PDF File size 0.20 MB Source: groups.csail.mit.edu
Useful Formulas for 6.838 Basic Geometry and Trigonometry Matrix Calculus A=1bh sin(θ ± π) = ±cosθ Checkoutmatrixcalculus.org for a handy matrix derivative calculation tool. The 2 2 Matrix Cookbook also contains a comprehensive list of identities. tanθ = sinθ cos(θ ± π) = ∓sinθ cosθ 2 cotθ = (tanθ)−1 sin(α ±β) = sinαcosβ ±cosαsinβ dY−1 = −Y−1dY Y−1 eA = P 1An cos2θ +sin2θ = 1 cos(α±β)=cosαcosβ∓sinαsinβ dt dt n n! iθ −1 sin(−θ) = −sinθ e =cosθ+isinθ ⊤ eABA =AeBA−1 cos(−θ) = cosθ d sint = −cost ∇x(x b) = b dt ∇ (a⊤Xb)=ab⊤ AeA =eAA d cost = sint X dt 1 ⊤ ⊤ ⊤ A B A+B+/2[A,B]+··· ∇x(x Ax+b x)=(A+A )x+b e e =e Linear Algebra d A ′ A p ∇Xtr(X)=I dte (t) = A (t)e (t) −1 −1 −1 (AB) =B A kAk = hA,Ai Fro ∇ tr(XB)=B⊤ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤ X (AB) =B A v · w = v w = tr(v w) = tr(wv ) P P tr(A) = n a = n λ kvk2 = v · v = v⊤v ∇ tr(X⊤BXC)=BXC+B⊤XC⊤ i=1 ii i=1 i 2 P X ⊤ p 1/p tr(A) = tr(A ) kvkp = ( i |vi| ) −⊤ Q tr(AB) = tr(BA) det(A) = n λ ∇Xdet(X)=det(X)·X P i=1 i ⊤ −1 1 hA,Bi= a b =tr(A B) det(A ) = /det(A) ij ij ij Differential Vector Calculus See this Wikipedia page for many vector calculus identities. dfx(v) = limh→0 f(x+hv)−f(x) = ∇f ·v ∇(φψ)=φ∇ψ+ψ∇φ h ∇f =(∂f ,..., ∂f ) ∇·(ψA)=ψ∇·A+(∇ψ)·A 1 n ∂x ∂x divF =∇·F =P ∂Fi ∇×(ψA)=ψ∇×A+(∇ψ)×A i ∂xi ∇·(∇×A)=0 curlF = ∇×F =( ∂ , ∂ , ∂ )×(Fx,Fy,Fz) for F : R3 → R3 ∂x ∂y ∂zP ∇×(∇×A)=∇(∇·A)+∆A ∆f =−∇2f =−∇·∇f =− n ∂2f i=1 ∂(xi)2 ∇×(∇ψ)=0 (in 6.838 we use a positive semidefinite Laplacian) (f ◦ g)′(t) = f′(g(t))g′(t) 1 ⊤ 3 f(x) = f(x )+∇f(x )·(x−x )+ (x−x ) Hf(x )(x−x )+O(kx−x k ) 0 0 0 2 0 0 0 0 2 Derivatives and Integrals, Integration by Parts, Stokes, etc. ! d ˆ b(t) f(x,t)dx =f(b(t),t)db(t) −f(a(t),t)da(t) +ˆ b(t) ∂f (x,t)dx ˆ (ψ∇·(ε∇φ)−φ∇·(ε∇ψ))dV =˛ ε(ψ∂φ −φ∂ψ)dA dt a(t) dt dt a(t) ∂t Ω ∂Ω ∂n ∂n d ˆ F(x,t)dV = ˆ ∂F(x,t)dV +˛ F(x,t)v ·nˆdA ˆ [G·(∇×F)−F ·(∇×G)]dV =˛ (F ×G)·nˆdA dt ∂t b D(t) D(t) Ω ∂Ω ˆ b ′ b ˆ b ′ ˆ G·∇fdV =˛ (fG)·nˆdA−ˆ f(∇·G)dV u(x)v (x)dx = [u(x)v(x)]a − u (x)v(x)dx Ω ∂Ω Ω ˆa u∇·V dA=˛ uV ·nˆdℓ−ˆ a∇u·V dA ˛ F · nˆ dA = ˆ ∇·FdV ∂Ω Ω ˆΩ ∂Ω ˛ Ω ˆ [F ·∇g+g(∇·F)]dV =˛ gF ·nˆ dA Ω(ψ∇·Γ+Γ·∇ψ)dA= ∂Ωψ(Γ·nˆ)dℓ Ω ∂Ω 1
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