Vector Field
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Scalar invariant
A scalar invariant is a scalar quantity that takes the same value in every coordinate system. The length (modulus) of a vector [Equation goes here - download the original pdf to see it.] is an example of a scalar invariant. This is a consequence of the next result Scalar products are scalar invariants Let P and Q be two points in Euclidean space [symbol] and let Oxyz and [Equation goes here - download the original pdf to see it.] be two orthonormal sets of coordinate axes such that the matrix [Equation goes here - download the original pdf to see it.] is the transformation matrix from Oxyz to [Equation goes here - download the original pdf to see it.] . Let the components of [Equation goes here - download the original pdf to see it.] relative to Oxyz be [Equation goes here - download the original pdf to see it.] and relative to [symbol] be [Equation goes here - download the original pdf to see it.] Then [Equation goes here - download the original pdf to see it.] Proof [Equation goes here - download the original pdf to see it.] The last line follows from the orthonormality conditions which follow from the identity for transformation matrices [Equation goes here - download the original pdf to see it.] In summation notation this is [Equation goes here - download the original pdf to see it.] This proof uses the fact that the orthonormality conditions are equivalent to the single statement [Equation goes here - download the original pdf to see it.] where [Equation goes here - download the original pdf to see it.] [Equation goes here - download the original pdf to see it.] is the Kronecker delta.. Given this result then [Equation goes here - download the original pdf to see it.] is scalar invariant; and hence [Equation goes here - download the original pdf to see it.] is scalar invariant.
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Contents of
Vector Field
1 Scalar and Vector Algebra -Prerequisites
2 Formal definition of a vector
3 The position vector
4 About vectors
5 Orthonormal basis
6 Scalar product
7 Scalar invariant
8 Resolute of a vector
9 The cross (vector) product
10 Distributive law
11 Basis vectors
12 Area of the parallelogram
13 Parallel, anti-parallel
14 The triple scalar product
15 The triple vector product
16 Quadruple vector products
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