Analysis (implicit and inverse func thrms)

In summary, the conversation discusses the equations uz - 2e^(vz) = 0, u - x^2 - y^2 = 0, and v^2 - xy*log(v) - 1 = 0, which define z as a function of (u,v) and (u,v) as a function of (x,y). The role of the inverse and implicit function theorems in this statement is to determine the Jacobian, specifically when it is equal to zero. This information is then used to compute the partial derivative of z with respect to x when x=0 and y=e, resulting in u=e^2, v=1, and z=2.
  • #1
steviet
3
0

Homework Statement



The equations:

uz - 2e^(vz) = 0
u - x^2 - y^2 = 0
v^2 - xy*log(v) - 1 = 0

define z (implicitly) as a function of (u,v) and (u,v) as a function of (x,y),
thus z as a function f (x,y)

Describe the role of the inverse and implicit function theorems in the
above statement and compute

[tex]\partial[/tex]z/[tex]\partial[/tex]x(0,e).

(Note that when x=0 and y=e, u=e^2, v=1 and z=2)

Homework Equations



Implicit and inverse function theorems

The Attempt at a Solution



I'm finally starting to get a grasp on the 2 theorems and their
respective proofs (I hope), but as far as explaining their role in this
concrete example I'm a little lost.
If anyone can put me on the right path, it would be greatly appreciated.
(This is for an undergrad analysis course)
 
Last edited:
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  • #2
Perhaps it would be a good idea to check on exactly what the "implicit function theorem" and ""inverse function theorem" say!

In particular you might want to determine where, if ever, the Jacobian is zero.
 

FAQ: Analysis (implicit and inverse func thrms)

1. What is implicit function theorem?

The implicit function theorem is a mathematical tool used to find solutions to equations that cannot be explicitly solved for a specific variable. It allows for the representation of one variable in terms of another variable without having to explicitly solve for the dependent variable.

2. How does the implicit function theorem work?

The implicit function theorem works by considering a system of equations and determining if there is a relationship between the variables that can be represented explicitly. If not, the theorem allows for the existence of a function that represents one variable in terms of the other.

3. What is the significance of the implicit function theorem?

The implicit function theorem is significant because it allows for the solution of equations that cannot be explicitly solved, making it a powerful tool in many areas of mathematics and science. It also allows for the understanding of the behavior of functions and their derivatives.

4. What is the inverse function theorem?

The inverse function theorem is a mathematical theorem that states that a function has an inverse if and only if its derivative is non-zero at a given point. It allows for the determination of the inverse of a function and the conditions under which it exists.

5. How is the inverse function theorem used in mathematics and science?

The inverse function theorem is used in many areas of mathematics and science, including calculus, differential equations, and physics. It allows for the determination of the inverse of a function, which is essential in solving many problems and understanding the behavior of functions and their derivatives.

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