Chemistry - Implicit function theorem

In mathematics, in the field of calculus of several variables, the implicit function theorem says that for a suitable set of equations, some of the variables are defined as a function of the others. There are some natural limitations on this use of a mathematical relation to define implicit functions, which may be seen in trying to use the unit circle as the graph of a function. Firstly, the projection of the circle onto the x-axis is two-to-one on the interval [−1,1]; this means that y can only be made a local function of x. Further at the points (1,0) and (−1,0), the tangent line to the circle is vertical. This means that y cannot there be a differentiable function of x.

The implicit function theorem gets round both these difficulties, which represent the typical obstructions. The implicit function is only locally defined, and points at which the first-order behaviour would be problematic are outside the scope of the result.

More precisely, that if

f:R^m+n→Rⁿ,

a is in R^m and b is in Rⁿ, with

f(a,b)=0;

if we take the Jacobian of f, and split it into submatrices:

$X=\left({\partial f_i \over \partial x_j}|_{(\mathbf{a},\mathbf{b})}\right)_{ij}$

$Y=\left({\partial f_i \over \partial y_j}|_{(\mathbf{a},\mathbf{b})}\right)_{ij}$

If Y is invertible (ie., the determinant Y is nonzero), f(x, y)=0 defines y as a function of x near (a,b), or that there exists a function such that g(b)=a, with its Jacobian at a being -Y^-1X.

Categories: Multivariate calculus | Theorems