# Vector Calculus

## Notation

$x = (x1, ⋯, xn).$$Dx ≡ (Dx1, ⋯, Dxn).$$grad(f) ≡ Dx(f) ≡ (Dx1(f), ⋯, Dxn(f)).$$div(f) ≡ Dx ⋅ f.$$curl(x) ≡ Dx × f.$$Δ(f) ≡ Dx2(f) ≡ (Dx ⋅ Dx)(f).$$Dx2(f) = Dx (Dx ⋅ f) − Dx × (Dx × f).$$∫L Dx t ⋅ dr = t(q) − t(p), for L = L[p → q].$$∫ ⋯ ∫V (Dx ⋅ f) dV = ∮ ⋯ ∮D V f ⋅ dS.$$∫∫Σ (Dx × f) ⋅ dΣ = ∮D Σ f ⋅ dr.$$∫∫A (Dx(M) − Dy(L)) dA = ∮D A (L dx + M dy).$

## Jacobian Matrix

$ji, j ≡ Dxj(fi).$

## Vector Calculus Identities

(Dx1, ⋯, Dxn) f = Dx1 f e1 + ⋯ + Dxn f en= Dxi f ei

DxB (a ⋅ b) = a × (Dx × b) + (a ⋅ Dx) b

### Distributive

$Dx(a + b) = Dx(a) + Dx(b).$$Dx ⋅ (a + b) = Dx ⋅ a + Dx ⋅ b.$$Dx × (a + b) = Dx × a + Dx × b.$

### Product Rule for Multiplication by a Scalar

$Dx(x y) = y Dx(x) + x Dx(y).$$Dx(x a) = (Dx(x))t a + x Dx(a) = Dx(x) ⊗ a + x Dx(a).$

Dx(x a) = x Dx ⋅ a = + (Dx x) ⋅ a Dx × (x a) = x Dx × + (Dx x) × a Dx2 (f g) = f Dx2 g + 2 Dx f ⋅ Dx g + g Dx2 f

### Quotient Rule for Division by a Scalar

Dx (x / y) = (y Dx x − x Dx y) / y2Dx(a / y) = (y Dx ⋅ a − Dx y ⋅ a) / y2Dx × (a / y) = (y Dx × a − Dx y × a) / y2

### Chain Rule

f : F → F, r(t) : Fn, g : Fn → F Dx (f ∘ g) = (f' ∘ g) Dx g(g ∘ r)' = (Dx g ∘ r) ⋅ r' Dx (g ∘ a) = (Dx g ∘ a) Dx a Dx(a ∘ y) = tr((Dx a ∘ y) jy)

### Dot Product Rule

Dx (a ⋅ b) = (a ⋅ Dx) b + (b ⋅ Dx) a + a × (Dx × b) + b × (Dx × a)
= a ⋅ jb + b ⋅ ja
= a ⋅ Dx b + b ⋅ Dx a

### Cross Product Rule

Dx(a × b) = (Dx × a) ⋅ b − a ⋅ (Dx × b)Dx × (a × b) = a (Dx ⋅ b) − b (Dx ⋅ a) + (b ⋅ Dx) a − (a ⋅ Dx) b
= (Dx ⋅ + b ⋅ Dx) a − (Dx ⋅ a + a ⋅ Dx) b
= Dx(b at) − Dx(a bt)
= Dx (b at − a bt)a × (Dx × b) = Dxb (a ⋅ b)(a ⋅ Dx) b
= a ⋅ jb(a ⋅ Dx) b
= a ⋅ Dx b − (a ⋅ Dx) b
= a ⋅ (jb(jb)t)(a × Dx) × b = a ⋅ Dx b − a (Dx ⋅ b)
a × (Dx × b) + (a ⋅ Dx) b − a (Dx ⋅ b)

∂Ω ω = ∫Ω dω.