Vector Calculus

Notation

x = (x_{1}, ⋯, x_{n}) .

D_{x} ≡ (D_{x_{1}}, ⋯, D_{x_{n}}) .

grad (f) ≡ D_{x} (f) ≡ (D_{x_{1}} (f), ⋯, D_{x_{n}} (f)) .

div (f) ≡ D_{x} ⋅ f.

curl (x) ≡ D_{x} × f.

Δ (f) ≡ {D_{x}}^{2} (f) ≡ (D_{x} ⋅ D_{x}) (f) .

{D_{x}}^{2} (f) = D_{x} (D_{x} ⋅ f) − D_{x} × (D_{x} × f) .

∫_{L} D_{x} t ⋅ dr = t (q) − t (p),   for L = L[p → q].

{∫ ⋯ ∫}_{V} (D_{x} ⋅ f) dV = {∮ ⋯ ∮}_{D V} f ⋅ dS.

{∫∫}_{Σ} (D_{x} × f) ⋅ dΣ = ∮_{D Σ} f ⋅ dr.

{∫∫}_{A} (D_{x} (M) − D_{y} (L)) dA = ∮_{D A} (L dx + M dy) .

Jacobian Matrix

j_{i, j} ≡ D_{x_{j}} (f_{i}) .

Vector Calculus Identities

(D_x₁, ⋯, D_{x_n}) f = D_x₁ f e₁ + ⋯ + D_{x_n} f e_n= D_{x_i} f e_i

D_x_B (a ⋅ b) = a × (D_x × b) + (a ⋅ D_x) b

Distributive

D_{x} (a + b) = D_{x} (a) + D_{x} (b) .

D_{x} ⋅ (a + b) = D_{x} ⋅ a + D_{x} ⋅ b.

D_{x} × (a + b) = D_{x} × a + D_{x} × b.

Product Rule for Multiplication by a Scalar

D_{x} (x y) = y D_{x} (x) + x D_{x} (y) .

D_{x} (x a) = {(D_{x} (x))}^{t} a + x D_{x} (a) = D_{x} (x) ⊗ a + x D_{x} (a) .

D_x ⋅ (x a) = x D_x ⋅ a = + (D_x x) ⋅ a D_x × (x a) = x D_x × + (D_x x) × a D_x² (f g) = f D_x² g + 2 D_x f ⋅ D_x g + g D_x² f

Quotient Rule for Division by a Scalar

D_x (x / y) = (y D_x x − x D_x y) / y²D_x ⋅ (a / y) = (y D_x ⋅ a − D_x y ⋅ a) / y²D_x × (a / y) = (y D_x × a − D_x y × a) / y²

Chain Rule

f : F → F, r(t) : F_n, g : Fⁿ → F D_x (f ∘ g) = (f' ∘ g) D_x g(g ∘ r)' = (D_x g ∘ r) ⋅ r' D_x (g ∘ a) = (D_x g ∘ a) D_x a D_x ⋅ (a ∘ y) = tr((D_x a ∘ y) j_y)

Dot Product Rule

D_x (a ⋅ b) = (a ⋅ D_x) b + (b ⋅ D_x) a + a × (D_x × b) + b × (D_x × a)
= a ⋅ j_b + b ⋅ j_a
= a ⋅ D_x b + b ⋅ D_x a

Cross Product Rule

D_x ⋅ (a × b) = (D_x × a) ⋅ b − a ⋅ (D_x × b)D_x × (a × b) = a (D_x ⋅ b) − b (D_x ⋅ a) + (b ⋅ D_x) a − (a ⋅ D_x) b
= (D_x ⋅ + b ⋅ D_x) a − (D_x ⋅ a + a ⋅ D_x) b
= D_x ⋅ (b a^t) − D_x ⋅ (a b^t)
= D_x (b a^t − a b^t)a × (D_x × b) = D_x_b (a ⋅ b) − (a ⋅ D_x) b
= a ⋅ j_b − (a ⋅ D_x) b
= a ⋅ D_x b − (a ⋅ D_x) b
= a ⋅ (j_b − (j_b)^t)(a × D_x) × b = a ⋅ D_x b − a (D_x ⋅ b)
a × (D_x × b) + (a ⋅ D_x) b − a (D_x ⋅ b)

Stokes' Theorem

∫_∂Ω ω = ∫_Ω dω.