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The Jacobian Matrix in the Implicit Function Theorem

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In the attached problem, I am having trouble showing that the determinants are in the same form in the Lagrangian as the one in the implicit function theorem.

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Let us start with a general discussion. Given a function G: R^nto R^m, denote by g_1,...,g_m: R^nto R the component
functions of G, meaning that for X=(x_1,...,x_n)in R^n,
Suppose that all the partial derivatives of g_1,...,g_m exist at a point {X}in R^n. We define the {Jacobian
matrix} of G at {X}, denoted by G_{{X}}({X}) and also by DG({X}) or G'({X}), by
the following expression:
pd{g_1({X})}{x_1} ... pd{g_1({X})}{x_n}
vdots ddots vdots
pd{g_m({X})}{x_1} ... pd{g_m({X})}{x_n}

The Jacobian matrix of a function from R^n to R^m is an mtimes n matrix, that is, it has m rows and n columns. The
number of rows equals the number component functions, or equivalently, equals the exponent of R where the image of G is
contained, in this case m; the number of columns equals the number of variables, in this case the vector {X} has n

Note that the i^{th}-row of the Jacobian matrix corresponds to the (transpose) of the gradient vector nabla g_i,
nabla g_i({X})={pmatrix}

so (nabla g_i)^T=(pd{g_i}{x_1},pd{g_i}{x_2},...,pd{g_i}{x_n}). Here the superscript T denotes transposition. We ...

Solution Summary

In the proof of the Theorem of Lagrange multipliers via the Implicit Function Theorem, we need to verify that a certain Jacobian matrix is non-singular. We explain how to calculate such a Jacobian matrix and why it is non-singular.

See Also This Related BrainMass Solution

Implicit Function Theorem and Theorem of Lagrange Multipliers

I don't get how they can take the derivative of g_1 and g_2 with respect to x_1 and x_2 when they are defined as
h_1=h_1 (x_3,x_4,...,x_n )=x_1 and h_2=h_2 (x_3,x_4,...,x_n )=x_2

I need a mathematical justification for how this can be written simply as (21) when x_1 and x_2 are defined as functions from other variables: x_3,x_4,...,x_n

Is it by using the chain rule or something else? Please go through the detailed steps for how this can be a normal quadratic matrix as defined in the implicit function theorem and by that showing why this is an applicable form of the implicit function theorem nonsingular matrix in (21):

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