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Summary of Differential Vector Operations in Various Coordinate Systems
Typology: Cheat Sheet
Uploaded on 03/11/2024
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Coordinate system
Gradient
@ x
e x
@ y
e y
@ z
e z
Divergence
ร ยฒ^ v ย
@v x
@ x
@v y
@ y
@v z
@ z
Curl
ร v ย
@v z
@ y
@v y
@ z
e x
@v x
@ z
@v z
@ x
e y
@v y
@ x
@v x
@ y
e (^) z
Laplacian of a scalar
2 T ย
2 T
@ x
2
2 T
@ y
2
2 T
@ z
2
Figure B.1 Unit vectors at the point ( x , y , z ).
Coordinate system
Gradient
@ r
e (^) r ย
r
@ q
e q
@ z
e (^) z (B-5)
Divergence
ร ยฒ v ย
r
@ r
r v r
r
@v q
@ q
@v z
@ z
Curl
ร v ย
r
@v z
@ q
@v q
@ z
e r
@v r
@ z
@v z
@ r
e q
r
@ r
r v q
@v r
@ q
e (^) z
Laplacian of a scalar
2 T ย
r
@ r
r
@ r
r
2
2 T
@ q
2
2 T
@ z
2
Coordinate system
Figure B.3 Unit vectors at the point ( r , q , f ).
Figure B.2 Unit vectors at the point ( r , q , z ).
704 Appendix B
4
(left column indicates 3D solids; right column indicates infinitely wide cross-sections)
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