Statement of The Theorem
Let f : Rn+m → Rm be a continuously differentiable function. We think of Rn+m as the Cartesian product Rn × Rm, and we write a point of this product as (x,y) = (x1, ..., xn, y1, ..., ym). Starting from the given function f, our goal is to construct a function g : Rn → Rm whose graph (x, g(x)) is precisely the set of all (x, y) such that f(x, y) = 0.
As noted above, this may not always be possible. We will therefore fix a point (a,b) = (a1, ..., an, b1, ..., bm) which satisfies f(a, b) = 0, and we will ask for a g that works near the point (a, b). In other words, we want an open set U of Rn, an open set V of Rm, and a function g : U → V such that the graph of g satisfies the relation f = 0 on U × V. In symbols,
To state the implicit function theorem, we need the Jacobian matrix of, which is the matrix of the partial derivatives of . Abbreviating (a1, ..., an, b1, ..., bm) to (a, b), the Jacobian matrix is
where is the matrix of partial derivatives in the 's and is the matrix of partial derivatives in the 's. The implicit function theorem says that if is an invertible matrix, then there are, and as desired. Writing all the hypotheses together gives the following statement.
- Let f : Rn+m → Rm be a continuously differentiable function, and let Rn+m have coordinates (x, y). Fix a point (a1,...,an,b1,...,bm) = (a,b) with f(a,b)=c, where c∈ Rm. If the matrix is invertible, then there exists an open set U containing a, an open set V containing b, and a unique continuously differentiable function g:U → V such that
Read more about this topic: Implicit Function Theorem
Famous quotes containing the words statement of the, statement of, statement and/or theorem:
“Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individuals sovereignty.”
—Mario Vargas Llosa (b. 1936)
“Truth is used to vitalize a statement rather than devitalize it. Truth implies more than a simple statement of fact. I dont have any whisky, may be a fact but it is not a truth.”
—William Burroughs (b. 1914)
“Children should know there are limits to family finances or they will confuse we cant afford that with they dont want me to have it. The first statement is a realistic and objective assessment of a situation, while the other carries an emotional message.”
—Jean Ross Peterson (20th century)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (19131960)