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Erschienen in:
2012 | OriginalPaper | Buchkapitel
Elementary Notation and Mathematical Operations
verfasst von : Tarek I. Zohdi
Erschienen in: Electromagnetic Properties of Multiphase Dielectrics
Verlag: Springer Berlin Heidelberg
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In this work, boldface symbols imply vectors or tensors.
A fixed Cartesian coordinate system will be used throughout this monograph.
The unit vectors for such a system are given by the (fixed) mutually orthogonal triad (
e
1
,
e
2
,
e
3
). For the inner product of two vectors,
u
and
v
, in three dimensions we have
$\bf u \cdot \bf v=\sum_{i=1}^3v_iu_i=u_1v_1+u_2v_2+u_3v_3=||\bf u|||\bf v||cos \theta,$
(2.1)
where
$||\bf u||=\sqrt{u_1^2+u_2^2+u_3^2}$
(2.2)
represents the Euclidean norm in
$I\!\!R^3$
and
θ
is the angle between them. We recall that a norm has three main characteristics for any two bounded vectors
u
and
v
(||
u
|| < ∞ and ||
v
|| < ∞):
||
u
|| ≥ 0, ||
u
|| = 0 if and only if
u
= 0,
||
u
+
v
|| ≤ ||
u
|| + ||
v
|| and
||
γ
u
|| = |
γ
|||
u
||, where
γ
is a scalar.