Transcription of Lectures on Vector Calculus - CSUSB
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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 .. 312 Vector Calculus Fields .. The Gradient .. Lagrange Multipliers .. The Divergence .. The Laplacian .. The Curl .. Vector Calculus with Indices .. Problems .. 473 Vector Calculus II: Other Coordinate Change of Variables from Cartesian to Spherical Polar.
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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