Properties of Coordinate Systems Cartesian Coordinates Position vector: r yy For Two Neighboring Points P and P : Displacement between two neighboring points: dsdr d dy y d Distance between two neighboring points (Found using the Pythagorean Theorem): ds dr d dy d
Primary Curve the curve obtained when one coordinate variable is allowed to vary while the other two are held fied. Primary Length Element infinitesimal length along the primary curve Primary Surface the surface obtained when the coordinate determining the primary length element is held fied and the other two are allowed to vary. Primary Element Primary Curve Primary Surface Primary Volume 1 st : nd : y 3 rd : Straight Line (-ais) (y and fied, varies) Straight Line (y-ais) ( and fied, y varies) Straight Line (-ais) ( and y fied, varies) y-plane ( fiedy and varies) -plane (y fied and varies) y-plane ( fied and y varies) Solid Cube Primary Length Elements Primary Area Elements Primary Volume Elements 1 st : d ( ) nd : dy ( ŷ ) 3 rd : d ( ẑ ) dy d ( ) d d ( ŷ ) d dy ( ẑ ) d dy d Primary length element vectors are in the direction of their corresponding primary curve. Primary area element vectors are in the same direction as the primary length element vector (i.e. to their corresponding primary surface). Primary volume elements are scalars not vectors and do not have an associated direction.
Coordinate Conversions Spherical Cartesian r sin cos y rsin sin rcos Unit Vector Conversions Spherical Cartesian r sin cos sinsiny cos θ cos cos cossiny sin φ sin cos y Cylindrical Cartesian cos y sin Cart Cylindrical Cartesian ρ cos sin y φ sin cos y Cart Special Functions Involving the Del Operator () f f f Gradient: f y y Divergence: F F F y y F y F F F y F F y F Curl: F y y y y F F F y Laplacian: f f f y f Note: f could also be a vector function F
Cylindrical Coordinates Position vector: r ρ or rcossin y For Two Neighboring Points P and P : Displacement between two neighboring points: dsdrd ρdφ d Distance between two neighboring points (Found using the Pythagorean Theorem): ds dr d d d
Primary Curve Primary Surface Primary Volume 1 st : Rays to the -ais ( and fied, varies) nd : Circle centered on the -ais ( and fied, varies) 3 rd : Straight line (-ais) ( and fied, varies) Cylinder centered on the -ais ( fied and varies) Half-plane from -ais ( fied and varies) Plane to the -ais ( fied and varies) Solid Cylinder Primary Length Elements Primary Area Elements Primary Volume Elements 1 st : d ( ρ ) dd ( ρ ) (teal surface) nd : d ( φ ) dd ( φ ) (purple surface) ddd 3 rd : d ( ẑ ) dd ( ẑ ) (pink surface) Coordinate Conversions Cartesian Cylindrical tan y y Cart y tan 1 Unit Vector Conversions Cartesian Cylindrical cos ρ sin φ y sin ρ cos φ Spherical Cylindrical r sin sph rcos Spherical Cylindrical r sin ρ cos θ cos ρ sin φ φ sph
Special Functions Involving the Del Operator () Gradient: f 1 f f f ρ φ 1 1 F F Divergence: F F 1 F F F F 1 F ρ φ Curl: F F Laplacian: 1 f 1 f f f
Spherical Coordinates Position vector: r r r or r rsin cos rsinsiny rcos For Two Neighboring Points P and P : Displacement between two neighboring points: dsdr drrrd θ rsindφ Distance between two neighboring points (Found using the Pythagorean Theorem): ds dr dr r d r sin d
Primary Element Primary Curve Primary Surface Primary Volume 1 st : r nd : 3 rd : Rays from the origin ( and fied, r varies) Half circle (r and fied, varies) Circle centered on polar ais (r and fied, varies) Sphere (r fied and varies) Cone of half angle ( fiedr and varies) Half-plane from -ais ( fied r and varies) Solid Sphere Primary Length Elements Primary Area Elements Primary Volume Elements 1 st : dr ( r ) r sindd ( r ) (teal) nd : r d ( θ ) r sindr d ( θ ) (pink) r sindr dd 3 rd : r sind ( φ ) r dr d ( φ ) (purple) Note: The r sin term is the distance from the polar ais to the projection of point P into the y-plane.
Coordinate Conversions Cartesian Spherical r y tan or tan y cos y y 1 tan cos y 1 y Cylindrical Spherical r tan or cos sph tan 1 cos 1 Unit Vector Conversions Cartesian Spherical sin cosr coscosθ sinφ y sin sinr cossinθ cosφ cos r sin θ Cylindrical Spherical ρ sin r cos θ φ φ sph cos r sin θ
Special Functions Involving the Del Operator () Gradient: 1 1 f f f r θ φ f r r r sin 1 1 1 F r r r sin r sin Divergence: F rfr sin F Curl: 1 F 1 1 F 1 F r r sin F rf rf F r sin r r sin r θ φ r r Laplacian: 1 1 1 f r f sin f f r r r r sin r sin