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The table displays 10 rows and 3 columns. The column headings from left to right are Cartesian coordinates, cylindrical coordinates and spherical coordinates. The table data is as follows:

Row 1. Coordinate variables. Cartesian coordinates, x, y, z. Cylindrical coordinates, r, phi, z. Spherical coordinates, R, theta, phi.

Row 2. Vector representation A equals. Cartesian coordinates, x caret A sub x plus y caret A sub y plus z caret A sub z. Cylindrical coordinates, r caret A sub r plus phi caret A sub phi plus z caret A sub z. Spherical coordinates, R caret A sub R plus theta caret A sub theta plus phi caret A sub phi.

Row 3. Magnitude of A, the absolute value of A equals. Cartesian coordinates, the square root of start expression A squared sub x plus A squared sub y plus A squared sub z end expression. Cylindrical coordinates, the square root of start expression A squared sub rplus A squared sub phi plus A squared sub z end expression. Spherical coordinates, the square root of start expression A squared sub R plus A squared sub Theta plus A squared sub phi end expression.

Row 4. Position vector, vector O P sub 1 equals. Cartesian coordinates, x caret x sub 1 plus y caret y sub 1 plus z caret z sub 1, for P (x sub 1, y sub 1 and z sub 1). Cylindrical coordinates, r caret r sub 1 plus z caret z sub 1 for P (r sub 1, phi sub 1, z sub 1). Spherical coordinates, R caret R sub 1 for P (R sub 1, theta sub 1, phi sub 1).

Row 5. Base vectors properties. Cartesian coordinates, x caret times x caret equals y caret times y caret equals z caret times z caret equals 1. X caret times y caret equals y caret times z caret equals z caret times x caret equals 0. x caret multiply y caret equals z caret. Y caret multiply z caret equals x caret. Z caret multiply x caret equals y caret. Cylindrical coordinates, r caret times r caret equals phi caret times phi caret equals z caret times z caret equals 1. R caret times phi caret equals phi caret times z caret equals z caret times r caret equals 0. r caret multiply phi caret equals z caret. Phi caret multiply z caret equals r caret. Z caret multiply r caret equals phi caret. Spherical coordinates, R caret times R caret equals theta caret times theta caret equals phi caret times phi caret equals 1. R caret times theta caret equals theta caret times phi caret equals phi caret times R caret equals 0. R caret multiply theta caret equals phi caret. Theta caret multiply phi caret equals R caret. Phi caret multiply R caret equals theta caret.

Row 6. Dot product, A times B equals. Cartesian coordinates, A sub x B sub x plus A sub y B sub y plus A sub z B sub z. Cylindrical coordinates, A sub r B sub r plus A sub phi B sub phi plus A sub z B sub z. spherical coordinates, A sub R B sub R plus A sub theta B sub theta plus A sub phi B sub phi.

Row 7. Cross product, A multiply B equals. Cartesian coordinates, a 3 by 3 matrix. Row 1. x caret, y caret, z caret. Row 2. A sub x, A sub y, A sub z. Row 3. B sub x, B sub y, B sub z. cylindrical coordinates, a 3 by 3 matrix. Row 1. r caret, phi caret, z caret. Row 2. A sub r, A sub phi, A sub z. Row 3. B sub r, B sub phi, B sub z. Spherical coordinates, a 3 by 3 matrix. Row 1. R caret, theta caret, phi caret. Row 2. A sub R, A sub theta, A sub phi. Row 3. B sub R, B sub theta, B sub phi.

Row 8. Differential length, d l equals. Cartesian coordinates, x caret d x plus y caret d y plus z caret d z. cylindrical coordinates, r caret d r plus phi caret r d phi plus z caret d z. spherical coordinates, upper R caret d R plus theta caret upper R d theta plus phi caret R sine of theta d phi.

Row 9. Differential surface areas. Cartesian coordinates, d s sub x equals x caret d y d z. d s sub y equals y caret d x d z. d s sub z equals z caret d x d y. Cylindrical coordinates, d s sub r equals r caret r d phi d z. d s sub phi equals phi caret d r d z. d s sub z equals z caret d r d phi. Spherical coordinates, d S sub upper R equals upper R caret R squared sine of theta d theta d phi. d s sub theta equals theta caret R sine of theta d R d phi. d s sub phi equals phi caret R d R d theta.

Row 10. Differential volume, d V equals. Cartesian coordinates, d x d y d z. Cylindrical coordinates, r d r d phi d z. spherical coordinates, upper R squared sine of theta d R d theta d phi.