Derivative Operators

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uv ∂f ∂f ∂f ∇ f = xˆ + yˆ + zˆ ∂x ∂y ∂z

Cartesian:

Cylindrical:

Spherical:

uv ∂f ˆ 1 ∂f ∂f ∇ f = ρˆ +θ + zˆ ∂ρ ρ ∂θ ∂z

uv ∂f 1 ∂f ˆ 1 ∂f ∇ f = rˆ + θˆ +φ ∂r r ∂θ r sin θ ∂φ

Divergence

uv uv ∂f ∂f ∂f ∇gf = x + y + z ∂x ∂y ∂z

Cartesian:

Cylindrical:

Spherical:

uv uv 1 ∂ 1 ∂fφ ∂f z ∇gf = ρ fρ ) + + ( ρ ∂ρ ρ ∂φ ∂z

uv uv 1 ∂ 1 ∂ 1 ∂fφ ∇gf = 2 ( r 2 f r ) + fθ sin θ ) + ( r ∂r r sin θ ∂θ r sin θ ∂φ

Laplacian Cartesian:

∇2 f =

∂ 2 f ∂ 2 f ∂2 f + + ∂x 2 ∂y 2 ∂z 2

2 Cylindrical: ∇ f =

Spherical:

∇2 f =

1 ∂ ∂f 1 ∂ 2 f ∂2 f ρ + 2 + ρ ∂ρ ∂ρ ρ ∂θ 2 ∂z 2

1 ∂ 2 ∂f 1 ∂  ∂f  1 ∂2 f r + sin θ +   r 2 ∂r ∂r r 2 sin θ ∂θ  ∂θ  r2 sin2 θ ∂φ2

1 ∂ 2 ∂f 1 ∂ 2 r = (rf ) Note that 2 r ∂r ∂r r ∂r 2

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