Transcription of Vectors and Index Notation - UCA
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Vectors and Index Notation - UCA
Vectors and Index NotationStephen R. AddisonJanuary 12, 20041 Basic Vector Unit VectorsWe will denote a unit vector with a superscript caret, thus adenotes a unit vector. a | a|=1If~xis a vector in thex-direction x=~x|~x|is a unit vector. We will usei,j,and k, or x, y,and z, ore1,e2and e3and a variety of variations without further Addition and SubtractionAddition and subtraction are depicted Scalar Products~a.~b=|~a||~b|cos 1~a.~b=~b.~asince cos =cos( ) Vector Products|~a ~b|=|~a||~b|sin ~a ~b= ~b ~a,Why?~c=~a ~b,~cis perpendicular to~aand~ Geometric InterpretationQPqArea of a triangle=12base perpendicular height=12|~Q||~P|sin SoA=12|~Q ~P|is the area of a triangle and, accordingly,|~Q ~P|=the area of a Area as a vectorAreas can also be expressed as vector quantities, for the parallelogram considered above, we
2.2 Summation convention and dummy indices Consider ~a.~b =axbx +ayby +azbz this indicates that we can write a scalar product as ~a.~b =aibi 5. In the term aibi, an index like i that is summed over is called a dummy index (or more cavalierly as a dummy variable). The index used is irrelevant - just as the integration variable is irrelevant in
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