Explore BrainMass

Explore BrainMass

    The Jacobian Matrix in the Implicit Function Theorem

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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.

    © BrainMass Inc. brainmass.com March 5, 2021, 1:32 am ad1c9bdddf


    Solution Preview

    Please look at the attached pdf file for a properly formatted answer.

    You can leave a message in this post if something needs clarification.

    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.