Vector operations in Various Coordinate SystemsEnvironmental Fluid Dynamics

1. Cartesian coordinates: (x, y, z)

Coordinate Symbol Velocity component Unit vector

x x u iy y v jz z w k

= ix

+ j

y+ k

z

=2

x2+2

y2+2

z2

v = ux

+v

y+w

z

v =

ux

uy

uz

vx

vy

vz

wx

wy

wz

v =

wy vzuz wxvx uy

2v = v = v =

2ux2 +

2uy2 +

2uz2

2vx2 +

2vy2 +

2vz2

2wx2 +

2wy2 +

2wz2

2. Cylindrical coordinates: (r, , z)

Coordinate Symbol Velocity component Unit vector

radial r u = vr iangle v = v jvertical z w = vz k

x = rcos, y = rsin, z = z

i = er =

cossin0

, j = e = sincos

0

, k = ez = 00

1

Let be a function of x, y, and z ((x, y, z)) and let be the transformation of on cylindricalpolar coordinates ((r, , z)). Then utilizing the chain rule:

x=

r

r

x+

x,

y=

r

r

y+

y,

z=

z

Knowing that r =x2 + y2 and using the equations for x and y above yields:

r

x=x

r= cos,

r

y=y

r= sin,

x=sinr

,

y=cos

r

Hence becomes:

=

r cos sinrr sin+

cosr

z

= r

cossin0

+ 1r

sincos0

+ z

001

=

rer +

1

r

e +

zez

The velocity components have to be rotated by the rotation matrix:

vxvyvz

= cos sin 0sin cos 0

0 0 1

vrv

vz

= vrcos vsinvrsin+ vcos

vz

Again applying the chain rule:

vxx

=vxr

r

x+vx

x,

vyy

=vyr

r

y+vy

y

which yields:

vxx

=

(vrr

cos vr

sin

)cos+

(vr

cos vrsin v

sin vcos)(sin

r

),

vyy

=

(vrr

sin+vr

cos

)sin+

(vr

sin+ vrcos+v

cos vsin)(cos

r

)and we obtain:

v = 1rvr +

vrr

+1

r

v

+vzz

=1

r

(rvr)

r+

1

r

v

+vzz

= ir

+ j1

r

+ k

z

=1

r

r

(r

r

)+

1

r22

2+2

z2

v = 1r

(rvr)

r+

1

r

v

+vzz

v =

vrr

1rvr

vrz

vr

1rv

vz

vzr

1rvz

vzz

v =

1r vz vz

vrz vzr

1rv +

vr 1r vr

2v = v = v =

1rr

(r vrr

)+ 1r2

2vr2 +

2vrz2

1rr

(rvr

)+ 1r2

2v2 +

2vz2

1rr

(r vzr

)+ 1r2

2vz2 +

2vzz2

3. Spherical coordinates: (r, , )

Coordinate Symbol Velocity component Unit vector

radial r w = vr ipolar angle v = v jlongitude u = v k

x = rsincos, y = rsinsin, z = rcos

i = er =

sincossinsincos

, j = e = coscoscossinsin

, k = e = sincos

0

The rotation matrix for spherical coordinates is:

vxvyvz

= sincos coscos sinsinsin sincos cos

cos sin 0

vrv

v

= vrsincos+ vcoscos vsinvrsinsin+ vsincos + vcos

vrcos vsin

You can now utilize the equations above to calculate the different operations on scalars andvectors yourself following the examples given for cylindrical polar coordinates.

= ir

+ j1

r

+ k

1

rsin

=2

r2+

2

r

r+

1

r22

2+

1

r2tan

+

1

r2sin2

2

2

v = 1r2

r

(r2vr

)+

1

rsin

(vsin) +

1

rsin

v

v =

1rsin( (vsin) v

)1

rsinvr 1r r (rv)

1rr (rv) 1r vr

2v = v = v =

2vrr2 +

2rvrr +

1r22vr2 +

1r2tan

vr +

1r2sin2

2vr2

2vr2 +

2rvr +

1r22v2 +

1r2tan

v +

1r2sin2

2v2

2vr2 +

2rvr +

1r22v2 +

1r2tan

v +

1r2sin2

2v2