State And Prove Implicit Function Theorem (2023)

FAQs

How do you prove the implicit function theorem? ›

Define the function φ1 : Rn−1 → R by φ1 (v) = y. The inductive step Assume the result of the Implicit Function Theorem holds for all appropriate g : Rn → Rm−1 whenever n ≥ m − 1. Let f : Rn → Rm, with n ≥ m. We wish to show that the result of the Implicit Function Theorem holds for f.

The implicit function theorem really just boils down to this: if I can write down m (sufficiently nice!) equations in n+m variables, then, near any sufficiently nice solution point, there is a function of n variables which give me the remaining m coordinates of nearby solution points.

The implicit function is generally of the form f(x, y)=0, or g(x, y, z)=0, and it contains all the variables, coefficients, constant on the left-hand side of the equation, and is equalized to zero.

If ∂R∂y ≠ 0, then R(x, y) = 0 defines an implicit function that is differentiable in some small enough neighbourhood of (a, b); in other words, there is a differentiable function f that is defined and differentiable in some neighbourhood of a, such that R(x, f(x)) = 0 for x in this neighbourhood.

The Implicit Function Theorem (discussed below) states that if ∂F/∂y = 0 at a point (x0,y0) in the solution set, then there is a neighbor- hood of (x0,y0) where the solution set is given by a function y = f(x).

Summary -- how to prove a theorem

Identify the assumptions and goals of the theorem. Understand the implications of each of the assumptions made. Translate them into mathematical definitions if you can. Make an assumption about what you are trying to prove and show that it leads to a proof or a contradiction.

We've “linearized” the function F, which we know is a good approximation to F in a neighborhood of (x, y), or of (∆x,∆y) = (0,0). The function F behaves like its linear approximation in this neighborhood, and the linear approximation behaves exactly like the linear function in Case 1.

Implicit Functions are different, in that x and y can be on the same side. A simple example is: xy = 1. It is here that implicit differentiation is used. Remember, you have used all of these derivative rules before.

An implicit function is a function of the form f(x, y) =0 that has been defined to aid in the differentiation of an algebraic function. The variables, coefficients, and constants are represented as an equation on the left-hand side of the implicit function, which has been equalized to zero.

The simplest example of an Implicit function theorem states that if F is smooth and if P is a point at which F,2 (that is, of/oy) does not vanish, then it is possible to express y as a function of x in a region containing this point.

Under what condition is the implicit function theorem not valid? ›

For a simple example of an equation that would not satisfy this, f(x,y)=y−⌊x⌋ is not continuously differentiable, so the implicit function theorem would fail for y−⌊x⌋=0.

You have to show f∘f(x)=x, and this is quite simple: f(f(x))=f(x||x||2)=x||x||2|x||x||2|2=x||x||2||x||2||x||4=x||x||4⋅||x||4=x. Done. f(f(x))=x|x|2|x|x|2|2=x|x|2×|x2|2|x|2=x.

Horizontal Line Test

Let f be a function. If any horizontal line intersects the graph of f more than once, then f does not have an inverse. If no horizontal line intersects the graph of f more than once, then f does have an inverse.