Page 493 - Maxwell House

APPENDIX 473

= cos + sin
0
0
0
= − sin + cos� (A.36)
0
0
0
= 0
0
= sincos + sinsin + cos
0
0
0
0
= coscos + cossin − sin� (A.37)
0
0
0
0
= − sin + cos
0
0
0
Note in conclusion that ∇ isn’t a real vector since vectors must have numbers or functions as
coordinates, but applying it correctly we can reserve plenty of our precious time for more
interesting tasks.
Vector Differential Operators in Cylindrical and Spherical Coordinate System
The cylindrical and spherical coordinate systems are mostly used in parallel with the Cartesian
system but not so wide. We tried to avoid them in our book as much as possible because of their
scary appearance and manipulation complexity but decided to keep here generally for
references.
Cylindrical coordinate system = + +
0
0
0
1
= 0 + 0 + 0 (A.38)

1 ( ) 1
∘ = + + (A.39)

1 1 ( ) � (A.40)
× = � − � + � − � + � −
0 0 0

2
2
1
1

2
∇ = � � + + (A.41)
2
2 2
2 2
2
2
2
2
grad(div) = ∇ = �∇ − − � + �∇ − + � + ∇
0

0

0
2
2
2 2
Spherical coordinate system = + +
0 0 0
1 1
= 0 + 0 + 0 (A.42)
sin
2
1 ( ) 1 (sin ) 1
∘ = + + (A.43)
2 sin sin
1 (sin ) 1 1 ( ) 1 ( )
× = 0 � − � + 0 � − � + 0 � − � (A.44)
sin sin
2 1 1
2
1 ()
2
∇ = + �sin � + (A.45)
2
2
2
2 sin sin 2
2 2 (sin ) 2

2
2
grad(div) = ∇ = �∇ − − − �

0
2
2
2 sin sin
2 2cos
2
+ �∇ − + − �
0 2 2 2 2
sin sin
2
+ �∇ − + 2 + 2cos � (A.46)

0
2
2
2
2
2
sin sin sin