Vector Calculus Formulas. Fundamental theorems (main result) Here, F(x, y, z) =
P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k. FT of Line Integrals: If F = ∇f, and the curve C ...
Vector Calculus Formulas Fundamental theorems (main result) Here, F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k. FT of Line Integrals:
If F = ∇ f , and the curve C has endpoints A and B, then Z F · dr = f (B) − f (A). C
"
! , ∂Q ∂P − F · dr dA = ∂x ∂y C
Green’s Theorem: D
(circulation-curl form)
,
"
F · dr,
∇ × F · n dσ =
Stokes’ Theorem:
where C is the edge curve of S
C
S
"
, ∇ · F dA =
Green’s Theorem: D
F · n ds
(flux-divergence form)
C
$ ∇ · F dV =
Divergence Theorem: D
F · n dσ S
Differential elements Along a parametrized curve r(t), t ∈ [a, b], we have ds = dr dt dt. ∂r × Along a parametrized surface r(u, v), (u, v) ∈ D, we have dσ = ∂u
∂r ∂v
du dv.
Curl and divergence For a continuously differentiable 3D vector field F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k, ! ! ! ∂Q ∂P ∂R ∂Q ∂P ∂R curl F = ∇ × F := − i+ − j+ − k. ∂y ∂z ∂z ∂x ∂x ∂y ! ∂ ∂ ∂ ∂P ∂Q ∂R div F = ∇ · F := , , · (P, Q, R) = + + . ∂x ∂y ∂z ∂x ∂y ∂z Other 3D spatial coordinates r =
p x2 + y2 = ρ sin φ, p ρ = x2 + y2 + z2 ,
x = r cos θ = ρ sin φ cos θ, y = r sin θ = ρ sin φ sin θ, z = ρ cos φ , dV = dz dy dx = r dz dr dθ = ρ2 sin φ dρ dφ dθ .
