NST IA Maths: Scalar & Vector Fields and their Integrals

18 terms

Position Vector x

(x,y,z)

(∂Φ/∂x, ∂Φ/∂y, ∂Φ/∂z)

∇Φ / |∇Φ|

Line integral of a scalar field

∫r Φ ds = ∫[t1,t2] Φ(x(s)) |dx/dt| dt

Line integral of a vector field

∫r F.dx = ∫[t1,t2] F(x(t)).dx/dt dt

Line integral of the gradient of a scalar field

∫r (∇Φ).dx , does not depend on the path taken between endpoints

Conservative Vector Field

F = -∇Φ
F.dx is an exact differential
for every pair of points the line integral ∫r F.dx is independent of the path taken
line integral for all closed curves ∫r F.dx = 0
curl F = ∇ x F = 0

∫s dS = ∫s ndS

dS = a²sinθdθdΦ

Unit Normal to Surface S

n = x/|x| = (sinθcosΦ, sinθsinΦ, cosθ)

Parametrisation of a curved surface

x = X(θ,Φ) = (asinθcosΦ, asinθsinΦ, acosθ)
dS = (∂X/∂θ X ∂X/∂Φ) dθdΦ

Flux

∫s F.dS = ∫s F.ndS

Spherical Vector Surface Element

dS = ndS = (sinθcosΦ, sinθsinΦ, cosθ)a²sinθdθdΦ

∇.F

Divergence Theorem (Gauss Theorem)

∫v (∇.F) dV = ∫s F.dS where V is a volume bounded by the closed surface S

Laplacian of Φ

div(grad Φ) = ∇²Φ = ∂²Φ/∂x² + ∂²Φ/∂y² + ∂²Φ/∂z²

∇ x F

Stokes' Theorem

∫s (∇ x F).dS = ∫c F.dx where S is an open surface bounded by the closed curve C