【やこびぎょうれつ (Jacobian matrix)】
多変数ベクトル値関数
f ( x ) = [ f 1 ( x 1 , ⋯ , x n ) ⋮ f m ( x 1 , ⋯ , x n ) ] {\displaystyle {\boldsymbol {f}}({\boldsymbol {x}})=\left[{\begin{array}{c}f_{1}(x_{1},\cdots ,x_{n})\\\vdots \\f_{m}(x_{1},\cdots ,x_{n})\end{array}}\right]}
を各変数に関して1階偏微分した1階反変1階共変テンソルのこと. 通常 J ( x ) {\displaystyle J({\boldsymbol {x}})} と行列で表記する:
J ( x ) := [ ∂ f 1 ∂ x 1 ( x ) ⋯ ∂ f 1 ∂ x n ( x ) ⋮ ⋮ ∂ f m ∂ x 1 ( x ) ⋯ ∂ f m ∂ x n ( x ) ] . {\displaystyle J({\boldsymbol {x}}):=\left[{\begin{array}{ccc}{\frac {\partial f_{1}}{\partial x_{1}}}({\boldsymbol {x}})&\cdots &{\frac {\partial f_{1}}{\partial x_{n}}}({\boldsymbol {x}})\\\vdots &&\vdots \\{\frac {\partial f_{m}}{\partial x_{1}}}({\boldsymbol {x}})&\cdots &{\frac {\partial f_{m}}{\partial x_{n}}}({\boldsymbol {x}})\end{array}}\right].}
ヤコビ行列の行列式をヤコビアンということもある.
![{\displaystyle {\boldsymbol {f}}({\boldsymbol {x}})=\left[{\begin{array}{c}f_{1}(x_{1},\cdots ,x_{n})\\\vdots \\f_{m}(x_{1},\cdots ,x_{n})\end{array}}\right]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/356d6f9e93376f6b6856557bc859f8227f128801)
と行列で表記する:
![{\displaystyle J({\boldsymbol {x}}):=\left[{\begin{array}{ccc}{\frac {\partial f_{1}}{\partial x_{1}}}({\boldsymbol {x}})&\cdots &{\frac {\partial f_{1}}{\partial x_{n}}}({\boldsymbol {x}})\\\vdots &&\vdots \\{\frac {\partial f_{m}}{\partial x_{1}}}({\boldsymbol {x}})&\cdots &{\frac {\partial f_{m}}{\partial x_{n}}}({\boldsymbol {x}})\end{array}}\right].}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/0b3b209556ad98b674ca261e55a7fa890e3d2350)