8/17/2019 Spherical Coord

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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 rememberthat expressions for the operations of vector analysis are  different   in different coordinates. Here wegive explicit formulae for cylindrical and spherical coordinates.

1 Cylindrical CoordinatesIn cylindrical coordinates,

x =  r cos φ , y = r sin φ , z  = z ,

we have

∇f  = r∂f ∂r

  + φ1r

∂f 

∂φ + z∂f 

∂z   ,

∇ · u = 1

r

∂ (rur)

∂r  +

 1

r

∂uφ

∂φ  +

 ∂uz

∂z   ,

∇× u = r1r ∂uz∂φ   −  ∂ uφ∂z   + φ∂ur∂z    −  ∂uz∂r  + z1r ∂ (ruφ)∂r   − ∂ur∂φ .2 Spherical Coordinates

In spherical coordinates,

x =  r sin θ cos φ, y = r sin θ sin φ, z  =  r cos θ ,

we have

∇f  = r∂f 

∂r

 + θ1

r

∂f 

∂θ

 + φ  1

r sin θ

∂f 

∂φ

  ,

∇ · u =  1

r2∂ (r2ur)

∂r  +

  1

r sin θ

∂ (uθ sin θ)

∂θ  +

  1

r sin θ

∂uφ

∂φ  ,

∇× u = r   1r sin θ

 ∂ 

∂θ(uφ sin θ) −

 ∂uθ

∂φ

 + θ1

r

  1

sin θ

∂ur

∂φ  −

  ∂ 

∂r(ruφ)

 + φ1

r

 ∂ 

∂r(ruθ) −

 ∂ur

∂θ

.