Translation for "sisätulon" to english

Sisätulon

• the inner product
• internal income

Translation examples

the inner product

Koska sisätulo on invariantti, näiden on oltava yhtä suuret:
Then since the inner product is an invariant, these must be equal:

Koska tämä on operaattori, sillä ei ole "pituutta", mutta jos lasketaan muodollisesti tämän operaattorin sisätulo itsensä kanssa, saadaan toinen operaattori:
Since this is an operator, it doesn't have a "length", but evaluating the inner product of the operator with itself gives another operator:

Yhtälöllä xTℓ = 0 lasketaan kahden pysty­vektorin sisätulo.
The equation xTℓ = 0 calculates the inner product of two column vectors.

Sisätulo on pistetulon yleistys abstarkteihin vektori­avaruuksiin, joiden kerroin­kuntana on joko reaalilukujen ( R {\displaystyle \mathbb {R} } ) tai kompleksilukujen kunta ( C {\displaystyle \mathbb {C} } ).
The inner product generalizes the dot product to abstract vector spaces over a field of scalars, being either the field of real numbers R {\displaystyle \mathbb {R} } or the field of complex numbers C {\displaystyle \mathbb {C} } .

Samoin kuin vektorien sisätulo lasketaan vastaavien komponenttien tulojen summana, funktion sisätulo määritellään integraalina jonkin välin a < x b yli, jolle välille käytetään myös merkintää : ⟨ u , v ⟩ = ∫ a b u ( x ) v ( x ) d x {\displaystyle \left\langle u,v\right\rangle =\int _{a}^{b}u(x)v(x)dx} Tämä voidaan edelleen yleistää kompleksifunktioille ψ ( x ) {\displaystyle \psi (x)} ja χ ( x ) {\displaystyle \chi (x)} samaan tapaan kuin kompleksivektorien sisätulo edellä määriteltiin.
This notion can be generalized to continuous functions: just as the inner product on vectors uses a sum over corresponding components, the inner product on functions is defined as an integral over some interval a ≤ x ≤ b (also denoted ): ⟨ u , v ⟩ = ∫ a b u ( x ) v ( x ) d x {\displaystyle \left\langle u,v\right\rangle =\int _{a}^{b}u(x)v(x)dx} Generalized further to complex functions ψ(x) and χ(x), by analogy with the complex inner product above, gives ⟨ ψ , χ ⟩ = ∫ a b ψ ( x ) χ ( x ) ¯ d x . {\displaystyle \left\langle \psi ,\chi \right\rangle =\int _{a}^{b}\psi (x){\overline {\chi (x)}}dx.} Inner products can have a weight function, i.e. a function which weights each term of the inner product with a value.

Tällöin kaikilla x ∈ H {\displaystyle x\in H} ∑ k = 1 ∞ | ⟨ x , e k ⟩ | 2 ≤ ‖ x ‖ 2 , {\displaystyle \sum _{k=1}^{\infty }\left\vert \left\langle x,e_{k}\right\rangle \right\vert ^{2}\leq \left\Vert x\right\Vert ^{2},} missä <∙,∙> on H:n sisätulo.
Then, for any x {\displaystyle x} in H {\displaystyle H} one has ∑ k = 1 ∞ | ⟨ x , e k ⟩ | 2 ≤ ‖ x ‖ 2 , {\displaystyle \sum _{k=1}^{\infty }\left\vert \left\langle x,e_{k}\right\rangle \right\vert ^{2}\leq \left\Vert x\right\Vert ^{2},} where 〈•,•〉 denotes the inner product in the Hilbert space H {\displaystyle H} .

Kanta voidaan esittää rivivektoreina: e 0 = ( 1 0 0 0 ) , e 1 = ( 0 1 0 0 ) , e 2 = ( 0 0 1 0 ) , e 3 = ( 0 0 0 1 ) {\displaystyle \mathbf {e} ^{0}={\begin{pmatrix}1&0&0&0\end{pmatrix}}\,,\quad \mathbf {e} ^{1}={\begin{pmatrix}0&1&0&0\end{pmatrix}}\,,\quad \mathbf {e} ^{2}={\begin{pmatrix}0&0&1&0\end{pmatrix}}\,,\quad \mathbf {e} ^{3}={\begin{pmatrix}0&0&0&1\end{pmatrix}}} niin että: A = ( A 0 A 1 A 2 A 3 ) {\displaystyle \mathbf {A} ={\begin{pmatrix}A_{0}&A_{1}&A_{2}&A_{3}\end{pmatrix}}} Perusteena näille merkintätavoille on se, että sisätulo on skalaari.
The bases can be represented by row vectors: E 0 = ( 1 0 0 0 ) , E 1 = ( 0 1 0 0 ) , E 2 = ( 0 0 1 0 ) , E 3 = ( 0 0 0 1 ) {\displaystyle \mathbf {E} ^{0}={\begin{pmatrix}1&0&0&0\end{pmatrix}}\,,\quad \mathbf {E} ^{1}={\begin{pmatrix}0&1&0&0\end{pmatrix}}\,,\quad \mathbf {E} ^{2}={\begin{pmatrix}0&0&1&0\end{pmatrix}}\,,\quad \mathbf {E} ^{3}={\begin{pmatrix}0&0&0&1\end{pmatrix}}} so that: A = ( A 0 A 1 A 2 A 3 ) {\displaystyle \mathbf {A} ={\begin{pmatrix}A_{0}&A_{1}&A_{2}&A_{3}\end{pmatrix}}} The motivation for the above conventions are that the inner product is a scalar, see below for details.

Massalliselle kappaleelle, jonka lepomassa tai invariantti massa on m', neliliikemäärä määritellään: P = m U = m γ ( u ) ( c , u ) = ( E / c , p ) {\displaystyle \mathbf {P} =m\mathbf {U} =m\gamma (\mathbf {u} )(c,\mathbf {u} )=(E/c,\mathbf {p} )} missä liikkuvan kappaleen kokonaisenergia on: E = γ ( u ) m c 2 {\displaystyle E=\gamma (\mathbf {u} )mc^{2}} ja sen relativistinen kokonaisliikemäärä on: p = γ ( u ) m u {\displaystyle \mathbf {p} =\gamma (\mathbf {u} )m\mathbf {u} } Laskemalla neliliikemäärän sisätulo itsensä kanssa saadaan: ‖ P ‖ 2 = P μ P μ = m 2 U μ U μ = m 2 c 2 {\displaystyle \|\mathbf {P} \|^{2}=P^{\mu }P_{\mu }=m^{2}U^{\mu }U_{\mu }=m^{2}c^{2}} ja myös: ‖ P ‖ 2 = E 2 c 2 − p ⋅ p {\displaystyle \|\mathbf {P} \|^{2}={\frac {E^{2}}{c^{2}}}-\mathbf {p} \cdot \mathbf {p} } mistä saadaan kappaleen energian ja liikemäärän välinen yhteys: E 2 = c 2 p ⋅ p + ( m c 2 ) 2 . {\displaystyle E^{2}=c^{2}\mathbf {p} \cdot \mathbf {p} +(mc^{2})^{2}\,.} Tämä tulos on käyttökelpoinen relati­visti­sessa meka­niikassa ja oleellisen tärkeä relati­visti­sessa kvantti­mekaniikassa sekä relati­visti­sessa kvantti­kenttä­teoriassa, joita kaikkia sovelletaan hiukkas­fysiikassa.
For a massive particle of rest mass (or invariant mass) m0, the four-momentum is given by: P = m 0 U = m 0 γ ( u ) ( c , u ) = ( E / c , p ) {\displaystyle \mathbf {P} =m_{0}\mathbf {U} =m_{0}\gamma (\mathbf {u} )(c,\mathbf {u} )=(E/c,\mathbf {p} )} where the total energy of the moving particle is: E = γ ( u ) m 0 c 2 {\displaystyle E=\gamma (\mathbf {u} )m_{0}c^{2}} and the total relativistic momentum is: p = γ ( u ) m 0 u {\displaystyle \mathbf {p} =\gamma (\mathbf {u} )m_{0}\mathbf {u} } Taking the inner product of the four-momentum with itself: ‖ P ‖ 2 = P μ P μ = m 0 2 U μ U μ = m 0 2 c 2 {\displaystyle \|\mathbf {P} \|^{2}=P^{\mu }P_{\mu }=m_{0}^{2}U^{\mu }U_{\mu }=m_{0}^{2}c^{2}} and also: ‖ P ‖ 2 = E 2 c 2 − p ⋅ p {\displaystyle \|\mathbf {P} \|^{2}={\frac {E^{2}}{c^{2}}}-\mathbf {p} \cdot \mathbf {p} } which leads to the energy–momentum relation: E 2 = c 2 p ⋅ p + ( m 0 c 2 ) 2 . {\displaystyle E^{2}=c^{2}\mathbf {p} \cdot \mathbf {p} +(m_{0}c^{2})^{2}\,.} This last relation is useful relativistic mechanics, essential in relativistic quantum mechanics and relativistic quantum field theory, all with applications to particle physics.