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Class_1 - PHYS809 Class 1 Notes 1 Vector Calculus 1.1...

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PHYS809Class1Notes 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
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1.4. Vector operator identities ( ) ( ) 2 0 0 . φ ∇ ∇× = ∇×∇ = ∇× ∇× = ∇ ∇ -∇ A A A A i i This last expression is useful for finding the Laplacian of a vector in non-Cartesian co-ordinates. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , , . ψ ψ ψ ψ ψ ψ = + ∇× = ∇ × + ∇× = + + × ∇× + × ∇× × = ∇× - ∇× ∇× × = - + - A A A A A A A B A B B A A B B A A B B A A B A B A B B A B A A B i i i i i i i i i i i i i 1.5. Grad, div and curl in general orthogonal curvilinear co-ordinates
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