Appendix

A.1 Summary of Vector Notation

The following write‐up concerning vector and tensor notation is developed from the work of many authors but I like the compendium by Bird et al. [1] the most.

A.1.1 Unit Vectors

Unit vectors are vectors of unit length in each of three orthogonal directions. The unit vectors are ModifyingAbove delta With right-arrow Subscript 1 Baseline comma ModifyingAbove delta With right-arrow Subscript 2 Baseline comma ModifyingAbove delta With right-arrow Subscript 3 Baseline. The following two properties summarize the dot and cross products of the unit vectors:

A.1-1StartLayout 1st Row ModifyingAbove delta With right-arrow Subscript i Baseline dot ModifyingAbove delta With right-arrow Subscript j Baseline equals delta Subscript italic i j Baseline equals StartLayout Enlarged left-brace 1st Row 1 if i equals j 2nd Row 0 if i not-equals j EndLayout 2nd Row ModifyingAbove delta With right-arrow Subscript i Baseline times ModifyingAbove delta With right-arrow Subscript j Baseline equals sigma-summation Underscript k equals 1 Overscript k equals 3 Endscripts epsilon Subscript italic i j k Baseline ModifyingAbove delta With right-arrow Subscript k Baseline 3rd Row where epsilon Subscript italic i j k Baseline equals StartLayout Enlarged left-brace 1st Row 1 if italic i j k equals 123 comma 231 comma 312 2nd Row negative 1 if italic i j k equals 321 comma 132 comma 213 3rd Row 0 if i equals j or comma i equals k or comma j equals k EndLayout EndLayout

Figure A.1-1 shows the directional characteristics of the three unit vectors.

A.1.2 Definition of a Vector

A vector is a quantity that has a magnitude and a direction associated with it. The vector υ can be written as:

A.1-2StartLayout 1st Row ModifyingAbove upsilon With right-arrow equals ModifyingAbove upsilon With right-arrow Subscript 1 Baseline plus ModifyingAbove upsilon With right-arrow Subscript 2 Baseline plus ModifyingAbove upsilon With right-arrow Subscript 3 Baseline equals ModifyingAbove delta With right-arrow Subscript 1 Baseline upsilon 1 plus ModifyingAbove delta With right-arrow Subscript 2 Baseline upsilon 2 plus ModifyingAbove delta With right-arrow Subscript 3 Baseline upsilon 3 equals sigma-summation Underscript i equals 1 Overscript 3 Endscripts ModifyingAbove delta With right-arrow Subscript i Baseline upsilon Subscript i EndLayout

where ModifyingAbove delta With right-arrow Subscript i are unit vectors oriented in the orthogonal directions and ModifyingAbove upsilon With right-arrow Subscript i are component vectors oriented in the orthogonal directions. υ i are the component vector magnitudes.

Schematic illustration of unit vectors in a Cartesian coordinate system.

Figure A.1-1 Unit vectors in a Cartesian coordinate system.

If the vector is ...

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