Translation for "partial derivatives" to finnish
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The relationship between real differentiability and complex differentiability is the following. If a complex function f(x + i y) = u(x, y) + i v(x, y) is holomorphic, then u and v have first partial derivatives with respect to x and y, and satisfy the Cauchy–Riemann equations:[6
Reaalisen ja kompleksisen derivoituvuuden välinen suhde voidaan esittää seuraavasti: Jos kompleksifunktio ƒ(x + i y) = u(x, y) + i v(x, y) on holomorfinen, funktioilla u ja v on ensimmäisen kertaluvun osittaisderivaatat x:n ja y:n suhteen.
We can now solve this by setting the partial derivatives to zero.
Ratkaisu löytyy asettamalla osittaisderivaatat nolliksi.
A simple converse is that if u and v have continuous first partial derivatives and satisfy the Cauchy–Riemann equations, then f is holomorphic.
Yksinkertaisimmillaan voidaan osoittaa, että jos funktioilla u ja v on jatkuvat ensimmäisen kertaluvun osittaisderivaatat, jotka toteuttavat Cauchyn–Riemannin yhtälöt, ƒ on holomorfinen.
The Jacobian matrix of f {\displaystyle f} is formed by the four partial derivatives of u {\displaystyle u} and v {\displaystyle v} with respect to x {\displaystyle x} and y {\displaystyle y} .
Tällöin f {\displaystyle \scriptstyle f} :n Jacobin matriisin muodostavat u {\displaystyle \scriptstyle u} :n ja v {\displaystyle \scriptstyle v} :n neljä osittaisderivaattaa x {\displaystyle \scriptstyle x} :n ja y {\displaystyle \scriptstyle y} :n suhteen.
A more satisfying converse, which is much harder to prove, is the Looman–Menchoff theorem: if f is continuous, u and v have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then f is holomorphic.
Tyydyttävämpi lause, joka on vaikeampi todistaa, on Looman–Menchoffin lause: jos funktio ƒ on jatkuva ja funktioilla u ja v on ensimmäisen kertaluvun jatkuvat osittaisderivaatat, jotka toteuttavat Cauchyn–Riemannin yhtälöt, ƒ on holomorfinen.
Using the standard basis, in index and abbreviated notations, the contravariant components are: ∂ = ( ∂ ∂ x 0 , − ∂ ∂ x 1 , − ∂ ∂ x 2 , − ∂ ∂ x 3 ) = ( ∂ 0 , − ∂ 1 , − ∂ 2 , − ∂ 3 ) = E 0 ∂ 0 − E 1 ∂ 1 − E 2 ∂ 2 − E 3 ∂ 3 = E 0 ∂ 0 − E i ∂ i = E α ∂ α = ( 1 c ∂ ∂ t , − ∇ ) = ( ∂ t c , − ∇ ) = E 0 1 c ∂ ∂ t − ∇ {\displaystyle {\begin{aligned}{\boldsymbol {\partial }}&=\left({\frac {\partial }{\partial x_{0}}},\,-{\frac {\partial }{\partial x_{1}}},\,-{\frac {\partial }{\partial x_{2}}},\,-{\frac {\partial }{\partial x_{3}}}\right)\\&=(\partial ^{0},\,-\partial ^{1},\,-\partial ^{2},\,-\partial ^{3})\\&=\mathbf {E} _{0}\partial ^{0}-\mathbf {E} _{1}\partial ^{1}-\mathbf {E} _{2}\partial ^{2}-\mathbf {E} _{3}\partial ^{3}\\&=\mathbf {E} _{0}\partial ^{0}-\mathbf {E} _{i}\partial ^{i}\\&=\mathbf {E} _{\alpha }\partial ^{\alpha }\\&=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},\,-\nabla \right)\\&=\left({\frac {\partial _{t}}{c}},-\nabla \right)\\&=\mathbf {E} _{0}{\frac {1}{c}}{\frac {\partial }{\partial t}}-\nabla \\\end{aligned}}} Note the basis vectors are placed in front of the components, to prevent confusion between taking the derivative of the basis vector, or simply indicating the partial derivative is a component of this four-vector.
Käytettäessä standardikantaa indeksi- ja lyhennysmerkintöineen neligradientin kontravariantit komponentit ovat: ∂ = ( ∂ ∂ x 0 , − ∂ ∂ x 1 , − ∂ ∂ x 2 , − ∂ ∂ x 3 ) = ( ∂ 0 , − ∂ 1 , − ∂ 2 , − ∂ 3 ) = e 0 ∂ 0 − e 1 ∂ 1 − e 2 ∂ 2 − e 3 ∂ 3 = e 0 ∂ 0 − e i ∂ i = e α ∂ α = ( 1 c ∂ ∂ t , − ∇ ) = e 0 1 c ∂ ∂ t − ∇ {\displaystyle {\begin{aligned}{\boldsymbol {\partial }}&=\left({\frac {\partial }{\partial x_{0}}},\,-{\frac {\partial }{\partial x_{1}}},\,-{\frac {\partial }{\partial x_{2}}},\,-{\frac {\partial }{\partial x_{3}}}\right)\\&=(\partial ^{0},\,-\partial ^{1},\,-\partial ^{2},\,-\partial ^{3})\\&=\mathbf {e} _{0}\partial ^{0}-\mathbf {e} _{1}\partial ^{1}-\mathbf {e} _{2}\partial ^{2}-\mathbf {e} _{3}\partial ^{3}\\&=\mathbf {e} _{0}\partial ^{0}-\mathbf {e} _{i}\partial ^{i}\\&=\mathbf {e} _{\alpha }\partial ^{\alpha }\\&=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},\,-\nabla \right)\\&=\mathbf {e} _{0}{\frac {1}{c}}{\frac {\partial }{\partial t}}-\nabla \\\end{aligned}}} On otettava huomioon, että kantavektorit on asetettu komponenttien, jotta tämä ei sekaantuisi kantavektorien derivaattoihin tai yksinkertaisesti sen osoittamiseksi, että osittaisderivaatta on tämän nelivektorin komponentti.
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