Week 6 Laplacian Operator

Question 1

what is the laplacian operator equation

Answer 1

∇^2 = ∇ * ∇

Question 2

how many terms are there for the laplacian of a scalar field and how many for a vector field

Answer 2

scalar has 3 terms

vector has 9 terms

Question 3

what are the three grad vector combination identities

Answer 3

∇ x ∇Φ = 0
∇ * (∇ x A) = 0
∇ x (∇ x A) = ∇ * (∇ * A) - ∇^2A

Question 4

what does the identity ∇ x ∇Φ = 0 mean in words

Answer 4

if a vector field F can be represented as the grad of a scalar field (∇Φ) then the curl of this vector field = 0 so is conservative/irrotational

Question 5

what does the identity ∇ * (∇ x A) = 0 mean in words

Answer 5

if a vector field F can be expressed as the curl of another vector field then the divergence of F = 0. Therefore F is a solenoidal field