“Jacobian”的概念、定义、翻译、参考文献-科学参考

Jacobian

Mathematics

For n functions f_i in n variables x_i, the Jacobian is the determinant of the Jacobian matrix. The absolute value of the Jacobian determinant is a measure of how much area or volume is being stretched or contracted locally. So, for small changes we have

$δ f_{1} δ f_{2} \dots δ f_{n} \approx | \frac{\partial (f_{1}, f_{2}, \dots, f_{n})}{\partial (x_{1}, x_{2},, \dots, x_{n})} | δ x_{1} δ x_{2} \dots δ x_{n} .$

For example, with the usual spherical polar coordinates r,θ‎,ϕ‎, the Jacobian determinant equals r²sinθ‎, showing that the same change in θ‎ contributes to a larger element of volume near this equator (θ‎ = π‎/2) compared to the north pole (θ‎ = 0), but the longitude φ‎ has no effect on the element of volume. See jacobian matrix.

单词	Jacobian
释义	Jacobian Mathematics For n functions f_i in n variables x_i, the Jacobian is the determinant of the Jacobian matrix. The absolute value of the Jacobian determinant is a measure of how much area or volume is being stretched or contracted locally. So, for small changes we have $δ f_{1} δ f_{2} \dots δ f_{n} \approx \| \frac{\partial (f_{1}, f_{2}, \dots, f_{n})}{\partial (x_{1}, x_{2},, \dots, x_{n})} \| δ x_{1} δ x_{2} \dots δ x_{n} .$ For example, with the usual spherical polar coordinates r,θ‎,ϕ‎, the Jacobian determinant equals r²sinθ‎, showing that the same change in θ‎ contributes to a larger element of volume near this equator (θ‎ = π‎/2) compared to the north pole (θ‎ = 0), but the longitude φ‎ has no effect on the element of volume. See jacobian matrix.
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