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...e = "vertical-align: top"> <math>\mathbb{R}^3</math></font> that preserve inner product (are [[isometric]]) and preserve orientation (have unit [[determinant]]). ...ter 9.10. Here it is proved that a linear operator on a finite-dimensional inner product space is normal if and only if it has a complete set of orthonormal eigenve

...\vec{a}</math> and <math>\vec{b}</math> in <math>\mathbb{R}^3</math> the [[inner product]] is invariant, that is, and observe that the inner product becomes by virtue of the orthonormality of the basis vectors

...e interaction is switched on, the symmetry of the atom is lowered to the [[inner product group]], which is isomorphic to SO(3) and is the subgroup of SO(3) &times; The inner product group consists of the following elements

...space structure; it requires imposing the additional structure given an [[inner product]] on the space. Compartmentalizing mathematical information in this way ca ...ll also be defined on the vector space, with the dot then representing the inner product between two vectors.

..., namely, as an affine space whose difference space is a three-dimensional inner product space. ====[[Normed space|Normed]], [[Banach space|Banach]], [[Inner product space|inner product]], and [[Hilbert space|Hilbert]] spaces====

...ment d''S'' and d'''''S''''' is of length d''S''. The dot stands for the [[inner product]] between the magnetic induction '''B''' and d'''''S'''''. In order to obtain the electromotive force (EMF) one must integrate the inner product of electric force with path, '''F'''_Lor&sdot;d'''l''', counter-

...'''k'''-space. We will show that divergence in '''r'''-space becomes an [[inner product]] in '''k'''-space and a curl becomes a [[cross product]]. Thus, we define

...esimal element and direction normal (perpendicular) to the element. The [[inner product]] <math>\scriptstyle \mathbf{J}\cdot d\mathbf{S}</math> is the current ''d

...up of transformations]] of two-dimensional complex vectors leaving their [[inner product]] fixed, and hence also the norm of a complex vector.<ref name= Mirman>

...as a real [[affine space]] whose difference space is a three-dimensional [[inner product space]]. For further details see [[Affine space#Euclidean space]] and [[spa ...s points, vectors and [[scalar product]] (called also [[dot product]] or [[inner product]]) of vectors without mentioning linear and affine spaces. Optionally, Cart

...as a real [[affine space]] whose difference space is a three-dimensional [[inner product space]]. For further details see [[Affine space#Euclidean space]] and [[spa ...s points, vectors and [[scalar product]] (called also [[dot product]] or [[inner product]]) of vectors without mentioning linear and affine spaces. Optionally, Cart

...(here <math> \scriptstyle \langle \cdot,\cdot\rangle </math> denotes the [[inner product]] between two vectors in the associated Hilbert space), and immediately aft

...''H''_''n''(''x'') are orthogonal in the sense of the following inner product:

...egrable functions on <math>\scriptstyle \mathbb{R}</math> with the complex inner product <math>\scriptstyle \langle f,g\rangle=\int_{-\infty}^{\infty}f(x)\overline{

Indeed, when we write the integral as an [[inner product]], it follows from partial integration and the vanishing of ''f''(''x'') o

The surface integral contains the [[inner product]] between the current density '''J''' (a vector) and a vector d'''S''' ort

...uld be, in general, ambiguous. For instance, in classical mechanics the [[inner product]] '''p'''&sdot;'''r''' is equal to '''r'''&sdot;'''p''', but simple replace ...radient operator '''&nabla;''' to the new coordinates and remember that an inner product involves, in general, a [[metric tensor]] '''g'''. Because the metric tenso

...''</sub> is the position of the ''i'' th particle; the dot stands for an [[inner product]] between the two 3-[[vector]]s. Indicate long-time averages by angular b

...ly, as an [[affine space]] whose difference space is a three-dimensional [[inner product space]]. For further details see [[Affine space#Euclidean space]] and [[spa

...ly, as an [[affine space]] whose difference space is a three-dimensional [[inner product space]]. For further details see [[Affine space#Euclidean space]] and [[spa