How Does the Implicit Function Theorem Apply to Close Values Around a Point?

In summary: ScientistIn summary, the Implicit function theorem is a mathematical tool used to solve equations that cannot be explicitly solved for a particular variable. In the example given, we see that for x very close to a, the value of y will also be very close to b, as long as b is not equal to 0. This is because when b is not equal to 0, the value of y will not be 0, which is when the partial derivative is 0. Additionally, for the solutions y=+sqrt(x-1) or y=-sqrt(x-1), only one will be valid for x very close to a, depending on the sign of b. This is because as x gets closer to a, the value of
  • #1
wakko101
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Hello all,

I've been reading through my textbook about the Implicit function theorem and have come across something in one the examples that I just don't understand:

Let F(x,y) = x - y^2 -1, then the partial der. F(a,b) WRT y = -2y, which is 0 only when b = 0. We know the possible solutions are y=+sqrt(x-1) or y=-sqrt(x-1).

This is the part that I don't understand:

"For x very close to a, only one of these solutions will be very close to b -- namely, sqrt(x-1) if b is greater than 0 and -sqrt(x-1) if b is less than 0 -- and this solution is the one that figues in the implicit function theorem."

I'm not sure how they arrive at this conclusion for "x very close to a".

Any help?

Cheers,
W.
 
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  • #2


I can provide some clarification on this topic. The Implicit function theorem is a mathematical tool used to solve equations that cannot be explicitly solved for a particular variable. In the example you mentioned, we have an equation F(x,y) = 0, and we are interested in finding a solution for y in terms of x (since y is the dependent variable).

Now, when we take the partial derivative of F with respect to y, we are essentially looking at how the value of y changes as x changes, while keeping all other variables constant. In this case, we can see that the partial derivative is -2y, which means that as x changes, y will change by a factor of -2. Therefore, for x very close to a, the value of y will also be very close to b, as long as b is not equal to 0. This is because when b is not equal to 0, the value of y will not be 0, which is when the partial derivative is 0.

Now, for the solutions y=+sqrt(x-1) or y=-sqrt(x-1), we can see that they are only valid when x-1 is positive, since we cannot take the square root of a negative number. Therefore, for x very close to a, only one of these solutions will be close to b, depending on the sign of b. This is because as x gets closer to a, the value of x-1 will also get closer to 0, and the only solution that will be valid is the one that has the same sign as b.

I hope this helps clarify the reasoning behind this conclusion. Let me know if you have any further questions. Keep up the good work in your studies!


 

FAQ: How Does the Implicit Function Theorem Apply to Close Values Around a Point?

What is the implicit function theorem?

The implicit function theorem is a mathematical tool used to find solutions to equations involving multiple variables. It states that if a function satisfies certain conditions, then one or more of its variables can be expressed as a function of the others.

What are the conditions for the implicit function theorem to apply?

The function must be continuously differentiable and satisfy a certain non-degeneracy condition, which ensures that the variables are not too dependent on each other. Additionally, the equation must be able to be written in the form F(x,y) = 0.

How is the implicit function theorem used in real-world applications?

The implicit function theorem has many applications in physics, engineering, and economics. It can be used to solve systems of equations, optimize functions, and model complex relationships between variables.

Can the implicit function theorem be applied to functions with more than two variables?

Yes, the implicit function theorem can be extended to functions with any number of variables. However, the equations become more complicated and the non-degeneracy condition becomes more difficult to satisfy as the number of variables increases.

Are there any limitations to the implicit function theorem?

Yes, the implicit function theorem has some limitations. It only applies to certain types of equations and may not always provide a unique solution. It also assumes that the function is differentiable, which may not always be the case in real-world situations.

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