Jacobian ideal

For the function V RN- R we shall calculate the Jacobian ideal. According to definition (A3)... 

In the Jacobian ideal of the function F(x) = xk there are no functions which would be proportional to monomials x, x2,...,xk 2 (or whose first non-zero terms of the Taylor expansion would contain these monomials). 

To establish which monomials in two variables, contained in Af2, are absent from the Jacobian ideal A(F), we shall represent the monomials belonging to M2 in the form of Pascal s triangle (this is a general procedure for computing the codimension of a function in two variables ... 

It follows from the form of Jacobian ideal and from the following equalities... 

Seemingly, the Jacobian ideal does not contain five monomials x, x2, y, y2, y3. Since the monomials x2, y3 are linearly dependent ... 

Calculating the Jacobian ideal of the following functions in one and two variables having a degenerate critical point x3, + x4, x5, x6, x7, x2y — y3, x3 + y3, (x2y + y4), x2y + y5, (x3 + y4) according to the rules presented in Section A2.2 we conclude that the above functions are /c-deter-mined, because their codimension is finite. Furthermore, on the basis of the form of Jacobian ideal we establish that the respective universal unfoldings of these functions have a form consistent with the functions compiled in Tables 2.2, 2.5 (in Section A2.2 we computed the Jacobian ideal, among other functions, for the functions x, x2, x3, x4,..., and for x3 + y3, x2y + y4). 

The smoothness of algebraic matrix groups is a property not shared by all closed sets in /c". To see what it means, take fc = fc and let 5 fc" be an arbitrary irreducible closed set. Let s be a point in S corresponding to the maximal ideal J in k[S]. If S is smooth, n si k = O si /J us) has fc-dimension equal to the dimension of S. (This would in general be called smoothness at s.) If S is defined by equations fj = 0, the generators and relations for OUS] show that S is smooth at s iff the matrix of partial derivatives (dfj/dXi)(s) has rank n — dim V. Over the real or complex field this is the standard Jacobian criterion for the solutions of the system (f = 0) to form a C or analytic submanifold near s. For S to be smooth means then that it has no cusps or self-crossings or other singularities . 

Prove that the rate of change of pressure with respect to volume is greater in magnitude along an adiabatic path relative to an isothermal path. Use the method of Jacobians and do not restrict your analysis to ideal gases. 

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