Vector Field
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Formal definition of a vector
vector is formally defined in three dimensions as a triad of scalars [Equation goes here - download the original pdf to see it.] that is invariant under translation of axes. That is, if the triad of scalars [Equation goes here - download the original pdf to see it.] referred to a set of axes Oxyz, and the axes are translated to [Equation goes here - download the original pdf to see it.] with triad [Equation goes here - download the original pdf to see it.] then [Equation goes here - download the original pdf to see it.]. Under a rotation of axes the vector [Equation goes here - download the original pdf to see it.] is transformed to the vector [Equation goes here - download the original pdf to see it.] by the transformation matrix [Equation goes here - download the original pdf to see it.] where the [symblol] are the direction cosines of the rotation of Oxyz to [Equation goes here - download the original pdf to see it.] . In summation notation this is written [Equation goes here - download the original pdf to see it.] For example [Equation goes here - download the original pdf to see it.] It can be shown that vectors, so defined, obey the axioms for vector algebra.
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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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