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Lectures on Vector Calculus - CSUSB

Lectures on Vector CalculusPaul RentelnDepartment of PhysicsCalifornia State UniversitySan Bernardino, CA 92407 March, 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 .. Determinants and Epsilon Symbols .. Vector Algebra III: Tensor Product .. Problems.

instead. (They are called ‘indices’ because they index something, and they are called ‘dummy’ because the exact letter used is irrelevant.) In index notation, then, I claim that the conditions (1.1) and (1.2) may be written e^ i^e j = ij: (1.3) How are we to understand this equation? Well, for starters, this equation

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1 Vector Algebra and Index Notation 1.1 Orthonormality and the Kronecker Delta We begin with three dimensional Euclidean space R 3.In R we can de ne

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