The Jacobian Matrix in the Implicit Function Theorem

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.

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

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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,
G({X})=(g_1({X}),...,g_m({X})).
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:
G_{{X}}({X})={pmatrix}
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}
{pmatrix}.

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
variables.

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}
pd{g_i({X})}{x_1}
pd{g_i({X})}{x_2}
vdots
pd{g_1({X})}{x_n}
{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.

Determine whether the equation xyz + sin(x + y + z) = 0 can be solved for z in a nonempty, open set V containing the point (x, y, z) = (0, 0, 0). If it can, is the solution differentiable near the point (x, y) = (0, 0)?
See the attached file. Solution to 11.6.2(a) is needed.

Real Analysis
Jacobians(I)
Necessary and sufficient condition for the value of a Jacobian of n independent functions to be zero
The fully formatted problem is in the attached file.

See attached file for thematrix.
(a) Calculate the characteristic polynomial Pa(t). Is a diagonalisable?
(b) State the Cayley-Hamilton theorem for a square matrix a.
(c) Using the Cayley-Hamilton theorem, compute a^5 with no more than one explitic matrix multiplication.

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

See the attached file as equations are contained within the Word file.
Consider the following function:
(see file)
where a > 0 is a parameter.
1. Find the first order condition for a critical point of this function.
2. Is this a maximum or a minimum or an inflection point?
3. Solve for x* (a), the maximizer of the fun

Use of ode45 to complete this...however, I am not sure how to go about it.
1.) integrate set of diff. equations with given initial conditions and constants and plot concentration profiles of all 5 dependent variables
2.) plot phase plot of s1 vs s2. What does the phase plot demonstrate? ...I don't know?
3.) perform a stabi

Assume that firm A produces good G using only labor. Therefore, the firm's output is a function of the quantity of labor hired (i.e. output = q(L)).
Assume further that this firm receives a price (p) for good G and pays laborers a wage (w) that are both constant, and that the firm pays a constant health care cost (h) for eac