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
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