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lim − ∞ → ∞ ∭ − ∞ ∞ { [ ∂ x ∂ y ∂ z f ( x ) ] d ( x ) , [ ∂ x ∂ y ∂ z f ( y ) ] d ( y ) , [ ∂ x ∂ y ∂ z f ( z ) ] d ( z ) } {\displaystyle \lim _{-\infty \to \infty }\iiint _{-\infty }^{\infty }{\begin{Bmatrix}{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(x)}}\end{bmatrix}}d(x),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(y)}}\end{bmatrix}}d(y),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(z)}}\end{bmatrix}}d(z)\end{Bmatrix}}}
[ R ] = [ lim − ∞ → ∞ ∭ − ∞ ∞ { [ ∂ x ∂ y ∂ z f ( x ) ] d ( x ) , [ ∂ x ∂ y ∂ z f ( y ) ] d ( y ) , [ ∂ x ∂ y ∂ z f ( z ) ] d ( z ) } lim − ∞ → ∞ ∭ − ∞ ∞ { [ ∂ x ∂ y ∂ z f ( x ) ] d ( x ) , [ ∂ x ∂ y ∂ z f ( y ) ] d ( y ) , [ ∂ x ∂ y ∂ z f ( z ) ] d ( z ) } lim − ∞ → ∞ ∭ − ∞ ∞ { [ ∂ x ∂ y ∂ z f ( x ) ] d ( x ) , [ ∂ x ∂ y ∂ z f ( y ) ] d ( y ) , [ ∂ x ∂ y ∂ z f ( z ) ] d ( z ) } ] {\displaystyle [\mathbf {R} ]={\begin{bmatrix}\lim _{-\infty \to \infty }\iiint _{-\infty }^{\infty }{\begin{Bmatrix}{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(x)}}\end{bmatrix}}d(x),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(y)}}\end{bmatrix}}d(y),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(z)}}\end{bmatrix}}d(z)\end{Bmatrix}}&&\\&\lim _{-\infty \to \infty }\iiint _{-\infty }^{\infty }{\begin{Bmatrix}{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(x)}}\end{bmatrix}}d(x),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(y)}}\end{bmatrix}}d(y),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(z)}}\end{bmatrix}}d(z)\end{Bmatrix}}&\\&&\lim _{-\infty \to \infty }\iiint _{-\infty }^{\infty }{\begin{Bmatrix}{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(x)}}\end{bmatrix}}d(x),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(y)}}\end{bmatrix}}d(y),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(z)}}\end{bmatrix}}d(z)\end{Bmatrix}}\end{bmatrix}}}
lim − ∞ → ∞ = [ f ( a + [ lim − ∞ → ∞ ∭ − ∞ ∞ { [ ∂ x ∂ y ∂ z f ( x ) ] d ( x ) , [ ∂ x ∂ y ∂ z f ( y ) ] d ( y ) , [ ∂ x ∂ y ∂ z f ( z ) ] d ( z ) } lim − ∞ → ∞ ∭ − ∞ ∞ { [ ∂ x ∂ y ∂ z f ( x ) ] d ( x ) , [ ∂ x ∂ y ∂ z f ( y ) ] d ( y ) , [ ∂ x ∂ y ∂ z f ( z ) ] d ( z ) } lim − ∞ → ∞ ∭ − ∞ ∞ { [ ∂ x ∂ y ∂ z f ( x ) ] d ( x ) , [ ∂ x ∂ y ∂ z f ( y ) ] d ( y ) , [ ∂ x ∂ y ∂ z f ( z ) ] d ( z ) } ] ) − f ( a ) [ lim − ∞ → ∞ ∭ ∞ ∞ { [ ∂ x ∂ y ∂ z f ( x ) ] d ( x ) , [ ∂ x ∂ y ∂ z f ( y ) ] d ( y ) , [ ∂ x ∂ y ∂ z f ( z ) ] d ( z ) } lim − ∞ → ∞ ∭ ∞ ∞ { [ ∂ x ∂ y ∂ z f ( x ) ] d ( x ) , [ ∂ x ∂ y ∂ z f ( y ) ] d ( y ) , [ ∂ x ∂ y ∂ z f ( z ) ] d ( z ) } lim − ∞ → ∞ ∭ ∞ ∞ { [ ∂ x ∂ y ∂ z f ( x ) ] d ( x ) , [ ∂ x ∂ y ∂ z f ( y ) ] d ( y ) , [ ∂ x ∂ y ∂ z f ( z ) ] d ( z ) } ] ] {\displaystyle \lim _{-\infty \to \infty }={\begin{bmatrix}{\frac {f(a+{\begin{bmatrix}\lim _{-\infty \to \infty }\iiint _{-\infty }^{\infty }{\begin{Bmatrix}{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(x)}}\end{bmatrix}}d(x),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(y)}}\end{bmatrix}}d(y),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(z)}}\end{bmatrix}}d(z)\end{Bmatrix}}&&\\&\lim _{-\infty \to \infty }\iiint _{-\infty }^{\infty }{\begin{Bmatrix}{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(x)}}\end{bmatrix}}d(x),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(y)}}\end{bmatrix}}d(y),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(z)}}\end{bmatrix}}d(z)\end{Bmatrix}}&\\&&\lim _{-\infty \to \infty }\iiint _{-\infty }^{\infty }{\begin{Bmatrix}{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(x)}}\end{bmatrix}}d(x),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(y)}}\end{bmatrix}}d(y),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(z)}}\end{bmatrix}}d(z)\end{Bmatrix}}\end{bmatrix}})-f(a)}{\begin{bmatrix}\lim _{-\infty \to \infty }\iiint _{\infty }^{\infty }{\begin{Bmatrix}{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(x)}}\end{bmatrix}}d(x),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(y)}}\end{bmatrix}}d(y),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(z)}}\end{bmatrix}}d(z)\end{Bmatrix}}&&\\&\lim _{-\infty \to \infty }\iiint _{\infty }^{\infty }{\begin{Bmatrix}{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(x)}}\end{bmatrix}}d(x),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(y)}}\end{bmatrix}}d(y),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(z)}}\end{bmatrix}}d(z)\end{Bmatrix}}&\\&&\lim _{-\infty \to \infty }\iiint _{\infty }^{\infty }{\begin{Bmatrix}{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(x)}}\end{bmatrix}}d(x),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(y)}}\end{bmatrix}}d(y),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(z)}}\end{bmatrix}}d(z)\end{Bmatrix}}\end{bmatrix}}}\end{bmatrix}}}
lim − ∞ → ∞ = { [ f a 1 ⋯ f p 1 ⋮ ⋱ ⋮ f a n ⋯ f p n ] + [ d x 1 ⋯ d p 1 ⋮ ⋱ ⋮ d x n ⋯ d p n ] } − { [ f a 1 ⋯ f p 1 ⋮ ⋱ ⋮ f a n ⋯ f p n ] } { [ d x 1 ⋯ d p 1 ⋮ ⋱ ⋮ d x n ⋯ d p n ] } {\displaystyle \lim _{-\infty \to \infty }={\frac {{\begin{Bmatrix}{\begin{bmatrix}fa_{1}&\cdots &fp_{1}\\\vdots &\ddots &\vdots \\fa_{n}&\cdots &fp_{n}\end{bmatrix}}+{\begin{bmatrix}dx_{1}&\cdots &dp_{1}\\\vdots &\ddots &\vdots \\dx_{n}&\cdots &dp_{n}\end{bmatrix}}\end{Bmatrix}}-{\begin{Bmatrix}{\begin{bmatrix}fa_{1}&\cdots &fp_{1}\\\vdots &\ddots &\vdots \\fa_{n}&\cdots &fp_{n}\end{bmatrix}}\end{Bmatrix}}}{\begin{Bmatrix}{\begin{bmatrix}dx_{1}&\cdots &dp_{1}\\\vdots &\ddots &\vdots \\dx_{n}&\cdots &dp_{n}\end{bmatrix}}\end{Bmatrix}}}}
f ( a ) = ∫ − ∞ ∞ f ( a ) e ∂ x ∂ y ∂ z d ( x ) {\displaystyle f(a)=\int _{-\infty }^{\infty }f(a)e^{\partial x\partial y\partial z}d(x)}
F ( A ) = ∭ − ∞ ∞ [ f ( a ) e ∂ x d ( x ) f ( b ) e ∂ y d ( y ) f ( c ) e ∂ z d ( z ) ] {\displaystyle F(A)=\iiint _{-\infty }^{\infty }{\begin{bmatrix}f(a)e^{\partial x}d(x)&&\\&f(b)e^{\partial y}d(y)&\\&&f(c)e^{\partial z}d(z)\\\end{bmatrix}}}
![{\displaystyle [\mathbf {R} ]={\begin{bmatrix}\lim _{-\infty \to \infty }\iiint _{-\infty }^{\infty }{\begin{Bmatrix}{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(x)}}\end{bmatrix}}d(x),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(y)}}\end{bmatrix}}d(y),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(z)}}\end{bmatrix}}d(z)\end{Bmatrix}}&&\\&\lim _{-\infty \to \infty }\iiint _{-\infty }^{\infty }{\begin{Bmatrix}{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(x)}}\end{bmatrix}}d(x),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(y)}}\end{bmatrix}}d(y),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(z)}}\end{bmatrix}}d(z)\end{Bmatrix}}&\\&&\lim _{-\infty \to \infty }\iiint _{-\infty }^{\infty }{\begin{Bmatrix}{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(x)}}\end{bmatrix}}d(x),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(y)}}\end{bmatrix}}d(y),{\begin{bmatrix}{\frac {\partial x\partial y\partial z}{f(z)}}\end{bmatrix}}d(z)\end{Bmatrix}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78fa5a728f28b9eae23003e11a7a6bdea31c2b85)
