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Unformatted text preview: PHYS809 Class 1 Notes 1. Vector Calculus 1.1. Gradient of a scalar field A scalar field ( ) φ r is a scalar function defined at all positions r . The change in φ in moving from r to r + d r is . d d φ φ = ∇ r i In Cartesian co-ordinates . x y z φ φ φ φ ∂ ∂ ∂ ∇ = + + ∂ ∂ ∂ i j k φ ∇ is a vector that lies in the direction of fastest increase in φ . It is normal to the surface φ = constant, that passes through r . 1.2. Divergence of a vector field In Cartesian co-ordinates . y x z A A A x y z ∂ ∂ ∂ ∇ = + + ∂ ∂ ∂ A i If 0, ∇ = A i A is said to be solenoidal . If , φ = ∇ A then 2 . φ φ ∇ = ∇ ∇ = ∇ A i i 2 ∇ is the Laplacian operator. 1.3. Curl of a vector field In Cartesian co-ordinates, curl . y y z x z x A A A A A A y z z x x y ∂ ∂ ∂ ∂ ∂ ∂ = ∇× =- +- +- ∂ ∂ ∂ ∂ ∂ ∂ A A i j k If 0, ∇× = A A is said to be irrotational . In solid body rotation, , = × v ω r where ω is fixed in magnitude and direction. Take ω = ω k , so that , y x ω ω = - + v i j and 2 2 . ω ∇× = = v k ω Hence if there is no rotation, 0. ∇× = v 1.4. Vector operator identities ( ) ( ) 2 . φ ∇ ∇× = ∇×∇ = ∇× ∇× = ∇ ∇-∇ A A A A i i This last expression is useful for finding the Laplacian of a vector in non-Cartesian co-ordinates....
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This note was uploaded on 12/02/2011 for the course PHYS 809 taught by Professor Macdonald during the Fall '11 term at University of Delaware.
- Fall '11