The Implicit Function Theorem implies that the system of equations u = x² - xy, v = 2xy - y² has a solution for x and y as functions of u and v valid in a neighbourhood of the point where x = y = 1.

Question

Quizplus · Accepted Answer

The Implicit Function Theorem states that if a system of equations can be locally expressed with continuously differentiable functions and the Jacobian matrix of the system (with respect to the variables we want to solve for) is non-singular at a point, then there exists a neighborhood around that point where solutions for the variables as functions of the others can be found. In this case, without explicitly calculating the Jacobian, the presence of x and y in both equations in a non-trivial manner (indicating dependence on x and y) and the assumption of differentiability (implicit in the context of the theorem's application) suggest that, at least near the point (x, y) = (1, 1), the system meets the criteria for the Implicit Function Theorem to guarantee local solutions for x and y as functions of u and v.

The Implicit Function Theorem Implies That the System of Equations