Grad Div and Curl in Cylindrical and Sph PDF

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differential eqn....

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MAS251/PHY202/MAS651

Handout 5

Grad, Div and Curl in Cylindrical and Spherical Coordinates In applications, we often use coordinates other than Cartesian coordinates. It is important to remember that expressions for the operations of vector analysis are different in different coordinates. Here we give explicit formulae for cylindrical and spherical coordinates.

1

Cylindrical Coordinates

In cylindrical coordinates, x = r cos φ , we have ∇f = rb

∇·u =

1 ∂uz ∂uφ b ∇×u = r − r ∂φ ∂z

2

y = r sin φ ,

z=z,

∂f ∂f b 1 ∂f + z b , +φ ∂z r ∂φ ∂r

1 ∂(rur ) 1 ∂uφ ∂uz , + + ∂z r ∂r r ∂φ

!

+

b φ

∂ur ∂uz − ∂z ∂r

!

"

#

1 ∂(ruφ ) ∂ur − + zb . ∂r r ∂φ

Spherical Coordinates

In spherical coordinates, x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ , we have

∇·u = ∇×u

= rb

"

∇f = rb

1 ∂f b 1 ∂f ∂f + θb +φ , r ∂θ ∂r r sin θ ∂φ

1 ∂(uθ sin θ) 1 ∂uφ 1 ∂(r 2 ur ) , + + 2 r sin θ r sin θ ∂φ ∂r ∂θ r #

"

#

"

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∂uθ ∂ ∂ur 1 1 ∂ ∂ 1 ∂ur b1 (ruθ ) − (uφ sin θ) − − (ruφ ) + φb +θ . sin θ ∂φ ∂r r sin θ ∂θ ∂φ r r ∂r ∂θ...