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Date: 23-9-2019
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Given a set of equations in variables , ..., , written explicitly as
(1) |
or more explicitly as
(2) |
the Jacobian matrix, sometimes simply called "the Jacobian" (Simon and Blume 1994) is defined by
(3) |
The determinant of is the Jacobian determinant (confusingly, often called "the Jacobian" as well) and is denoted
(4) |
The Jacobian matrix and determinant can be computed in the Wolfram Language using
JacobianMatrix[f_List?VectorQ, x_List] :=
Outer[D, f, x] /; Equal @@ (Dimensions /@ {f, x})
JacobianDeterminant[f_List?VectorQ, x_List] :=
Det[JacobianMatrix[f, x]] /;
Equal @@ (Dimensions /@ {f, x})
Taking the differential
(5) |
shows that is the determinant of the matrix , and therefore gives the ratios of -dimensional volumes (contents) in and ,
(6) |
It therefore appears, for example, in the change of variables theorem.
The concept of the Jacobian can also be applied to functions in more than variables. For example, considering and , the Jacobians
(7) |
|||
(8) |
can be defined (Kaplan 1984, p. 99).
For the case of variables, the Jacobian takes the special form
(9) |
where is the dot product and is the cross product, which can be expanded to give
(10) |
REFERENCES:
Gradshteyn, I. S. and Ryzhik, I. M. "Jacobian Determinant." §14.313 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1068-1069, 2000.
Kaplan, W. Advanced Calculus, 3rd ed. Reading, MA: Addison-Wesley, pp. 98-99, 123, and 238-245, 1984.
Simon, C. P. and Blume, L. E. Mathematics for Economists. New York: W. W. Norton, 1994.
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"آبل" تشغّل نظامها الجديد للذكاء الاصطناعي على أجهزتها
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تستخدم لأول مرة... مستشفى الإمام زين العابدين (ع) التابع للعتبة الحسينية يعتمد تقنيات حديثة في تثبيت الكسور المعقدة
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