NST IA Maths: Scalar & Vector Fields and their Integrals
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18 terms
Terms | Definitions |
|---|---|
 Position Vector x | (x,y,z) |
| grad Φ = ∇Φ | (∂Φ/∂x, ∂Φ/∂y, ∂Φ/∂z) |
| Unit normal to a surface | ∇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 |
| Vector Area S | ∫s dS = ∫s ndS |
| Spherical Scalar Area Element | 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 |
| Divergence, div F | ∇.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 |
| Curl F | ∇ x F |
| Stokes' Theorem | ∫s (∇ x F).dS = ∫c F.dx where S is an open surface bounded by the closed curve C |
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