Lectures on Vector Calculus - CSUSB
Lectures on Vector CalculusPaul RentelnDepartment of PhysicsCalifornia State UniversitySan Bernardino, CA 92407March, 2009; Revised March, 2011c Paul Renteln, 2009, 2011iiContents1 Vector Algebra and Index Orthonormality and the Kronecker delta . . . . . . . . . . . . Vector Components and Dummy Indices . . . . . . . . . . . . Vector Algebra I: Dot Product . . . . . . . . . . . . . . . . . . The Einstein Summation Convention . . . . . . . . . . . . . . Dot Products and Lengths . . . . . . . . . . . . . . . . . . . . Dot Products and Angles . . . . . . . . . . . . . . . . . . . . . Angles, Rotations, and Matrices . . . . . . . . . . . . . . . . . Vector Algebra II: Cross Products and the Levi Civita Symbol Products of Epsilon Symbols.
The Kronecker delta assumes nine possible values, depending on the choices for iand j. For example, if i = 1 and j = 2 we have 12 = 0, because iand jare not equal. If i= 2 and j= 2, then we get 22 = 1, and so on. A convenient way of remembering the de nition (1.6) is to imagine the Kronecker delta as a 3 by 3 matrix, where the rst index ...
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