Implicit function theorem, theory help.

In summary, the conversation discusses the implicit function theorem and its application in finding y as a function of x. The speaker questions why we can't be at a specific point when using the theorem and provides an example equation. The conversation also explores the reason for using the theorem when the equation is defined explicitly.
  • #1
Kruum
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0
For couple of days now I've been trying to figure out the implicit function theorem. Especially why we have to be near a point, why can't we be exactly in the point when trying to find out y as a function of x.

Let's take an easy equation as an example. Find [tex]y=y(x)[/tex] when [tex]f(x,y)=x^2+y^2-1[/tex]. If I solve this for y, it gives me [tex]y=\sqrt{1-x^2}[/tex] and [tex]y=-\sqrt{1-x^2}[/tex]. Now there shouldn't be any reasons for why I can't calculate y, when x=1. But if I use the implicit function theorem, it says you can't define y=y(x) at x=0, because [tex]f_y(x,y(x))=0[/tex].

Now ignore the fact that the function in the example is defined explicitly, think we had an implicit function that behaved like the one here. Can anybody explain me why is this?
 
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  • #2
Why can't we define the function at x=1? What is the reason behind using the implicit function theorem when the equation is defined explicitly.
 

Related to Implicit function theorem, theory help.

What is the Implicit Function Theorem?

The Implicit Function Theorem is a mathematical theorem that states under certain conditions, a system of equations can be represented as a differentiable function. This function can then be used to solve for one of the variables in terms of the other variables.

What are the conditions for the Implicit Function Theorem to hold?

The conditions for the Implicit Function Theorem to hold are: the system of equations must be differentiable, the equations must be independent, and the partial derivative of one of the variables with respect to another must be nonzero at the point of interest.

What is the significance of the Implicit Function Theorem in mathematics?

The Implicit Function Theorem is an important tool in mathematics, particularly in the fields of calculus and differential equations. It allows for the solution of equations that cannot be solved explicitly, and it is also used in optimization problems and in the study of curves and surfaces.

Can the Implicit Function Theorem be applied to any type of equation?

No, the Implicit Function Theorem can only be applied to a system of equations that meet the conditions stated in the theorem. These conditions must be checked for each individual case before applying the theorem.

How is the Implicit Function Theorem used in real-world applications?

The Implicit Function Theorem has many real-world applications, such as in economics, physics, and engineering. It can be used to model and solve various systems and phenomena, such as population growth, heat transfer, and electrical circuits.

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