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The table displays 6 rows and 4 columns. The column headings from left to right are transformation, coordinate variables, unit vectors, vector components. The table data is as follows:

Row 1. Transformation, Cartesian to cylindrical. Coordinate variables, r equals the square root of start expression x squared plus y squared end expression, phi equals tangent to the power negative left parenthesis y over x end expression. z equals z. Unit vectors, r caret equals x caret cosine phi plus y caret sine phi. phi caret equals negative x caret sin phi plus y caret cosine phi. z caret equals z caret. Vector components, A sub r equals A sub x cosine phi plus A sub y sine phi. A sub phi equals negative A sub x sine phi plus A sub y cosine phi. A sub z equals A sub z.

Row 2. Transformation, Cartesian to Cartesian. Coordinate variables, x equals r cosine phi. y equals r sine phi. z equals z. unit vectors, x caret equals r caret cosine phi minus phi caret sine phi. y caret equals r caret sine phi plus phi cosine phi. z caret equals z caret. Vector components, A sub x equals A sub r cosine phi minus A sub phi sine phi. A sub y equals A sub r sine phi plus A sub phi cosine phi. A sub z equals A sub z.

Row 3. Transformation, Cartesian to spherical. Coordinate variables, upper R equals the square root of start expression x squared plus y squared plus z squared end expression. Theta equals tangent to the power negative 1 left bracket the square root of start expression x squared plus y squared end expression over z right bracket. Phi equals tangent to the power negative 1 of y over x. Unit vectors, R caret equals x caret sine theta cosine phi plus y caret sin theta sin phi minus z caret sin theta. Phi caret equals negative x caret sine phi plus y caret cosine phi. Vector components, A sub Upper R equals A sub x sine theta cosine phi plus A sub y sine theta sine phi plus A sub z cosine theta. A sub theta equals A sub x cosine theta cosine phi plus A sub y cosine theta sine phi minus A sub z sine theta. A sub phi equals negative A sub x sine phi plus A sub y cosine phi.

Row 4. Transformation, Spherical to Cartesian. Coordinate variable, x equals R sine theta cosine phi. y equals R sine theta sine phi. z equals R cosine theta. Unit vectors, x caret equals R caret sin theta cosine phi plus theta caret cosine theta cosine phi minus phi caret sin phi. y caret equals R caret sin theta sin phi plus theta caret cosine theta sine phi plus phi caret cosine phi. z caret equals R caret cosine theta minus theta caret sine theta. Vector components, A sub x equals A sub upper R sine theta cosine phi plus A sub theta cosine theta cosine phi minus A sub phi sine phi. A sub y equals A sub R sine theta sine phi plus A sub theta cosine theta sine phi plus A sub phi cosine phi. A sub z equals A sub R cosine theta minus A sub theta sine theta.

Row 5. Transformation, cylindrical to spherical. Coordinate variable, R equals the square root of start expression r squared plus z squared end expression. Theta equals tangent to the power negative 1 r over z. phi equals phi. Unit vectors, upper R caret equals r caret sine theta plus z caret cosine theta. Theta caret equals r caret cosine theta minus z caret sine theta. Phi caret equals phi caret. Vector components, A sub upper R equals A sub r sine theta plus A sub z cosine theta. A sub theta equals A sub r cosine theta minus A sub z sine theta. A sub phi equals A sub phi.

Row 6. Transformation, spherical to cylindrical. Coordinate variable, r equals upper R sine theta. Phi equals phi. z equals R cosine theta. Unit vectors, R caret equals r caret sine theta plus z caret cosine theta. Theta caret equals r caret cosine theta minus z caret sine theta. Phi caret equals phi caret. Vector components. A sub r equals A sub upper R sine theta plus A sub theta cosines theta. A sub phi equals A sub phi. A sub z equals A sub upper R cosine theta minus A sub theta sine theta.