Inverse of multivariable mappings

In summary, the necessary conditions for an inverse for a multivariable mapping are that the mapping must be one-to-one and onto. For Euclidean spaces, this means that the spaces must have the same dimension. In the case of a mapping from a two-dimensional space to a one-dimensional space, the mapping is not one-to-one and therefore not invertible. Additionally, for a function with n independent variables, the dimension of the space it maps into cannot exceed n. Therefore, a mapping from R^n to R^x, where x is not equal to n, is not invertible.
  • #1
Gear300
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If there was a function f(x,y) = z...would the inverse be defined under injection (one-to-one correspondence) only if this condition was held?: Every z corresponds to only one (x,y) pair.

If this was the case...then wouldn't that imply that a good number of 2-D surfaces cannot have an inverse mapping?
Furthermore...what multivariable linear mapping would have an inverse (its not too hard to imagine multiple different orders of variables for which the same value is designated)?

Edit: my question is more or less this: What are the necessary conditions for an inverse for a multivariable mapping? If one-to-one correspondence is one of them, then what is the definition of one-to-one correspondence for multivariable mappings?
 
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  • #2
Good question: the following ideas may help.




Let g: A -> B be a function from a set A into a set B.

Definitions of one-to-one and onto:

g is one-to-one iff for every a1, a2 contained in A, g(a1) = g(a2) implies that a1 = a2.

g is onto iff for every b contained in B, there exists at least one a contained in A such that
g(a) = b.

Let X, and Y be sets, and f:X ->Y be a function from X to Y. Then f has an inverse iff f is one-to-one and onto.



It turns out that for Euclidean spaces R^n, n >=1 n is the dimension of the space (n = 2, are two dimensional planes, for n = 3, you are looking at three dimensional flat space), like the ones you mentioned below, there exists a one-to-one and onto mapping between the spaces iff the spaces have the same dimension. Your mapping that you are looking at is a mapping from a two dimensional Euclidean space to a one dimensional Euclidean space, and therefore can not be one-to-one, and so can not be invertible.

The proof to the theorems above can be found readily via a quick google search, or probably somewhere on this website: it is a really good website for mathematics and physics, and other subjects in the sciences :)


Also, another way to think about this particular kind of problem, that may be quite intuitive:

if you have a function from the complex numbers into a vector space of the form
w = f(x1, x2, ..., xn), where where x1, x2, ... xn are independant variables are allowed to vary over a set of real or complex values in some subset A of the complex numbers, then
f(A) can have dimension at most equal to n. Functions of n independant variables map into spaces of dimension less than or equal to n.

Another general principle that usually holds: if you have n unknown parameter values in a system, you need at least n equations, conditions,...etc., to specify a unique state of the system.


Not everything above was stated precisely, they are just ways of thinking about the ideas: which ideas apply depend on the particulars of the concepts you are considering. Hope this helps.
 
  • #3
Thanks. I thought this particular thread would die out, but a reply came (a good one). But to be clear, is what you were stating this?: Any mapping from Rn to Rx, x not equal to n, is not invertible.
 

FAQ: Inverse of multivariable mappings

What is the definition of the inverse of a multivariable mapping?

The inverse of a multivariable mapping is a function that undoes the original mapping. It takes the output from the original function and returns the original input. In other words, if f(x,y) is a multivariable function, its inverse, denoted as f-1(x,y), will result in the original input (x,y) when applied to the output of f(x,y).

How do you find the inverse of a multivariable mapping?

The process of finding the inverse of a multivariable mapping involves solving for the input variables in terms of the output variables. This can be done by using algebraic techniques such as substitution, elimination, or matrix operations. It is important to note that not all multivariable mappings have inverses, and the existence of an inverse depends on the specific properties of the mapping.

What is the importance of the inverse of a multivariable mapping?

The inverse of a multivariable mapping is important in many areas of mathematics, physics, and engineering. It allows us to undo a given mapping and retrieve the original input, which is useful in solving equations, optimization problems, and understanding the behavior of complex systems. Inverse mappings also play a crucial role in the study of calculus, differential equations, and linear algebra.

Can a multivariable mapping have more than one inverse?

No, a multivariable mapping can only have one inverse. This is because the inverse function must be a true function, meaning that each input corresponds to exactly one output. If a mapping has multiple inverses, it would break this definition of a function. However, a multivariable mapping can have different domains and ranges, resulting in different inverse functions.

What is the relationship between the inverse of a multivariable mapping and its derivative?

The inverse of a multivariable mapping and its derivative have an inverse relationship. This means that if the original function has a derivative at a point, then its inverse will also have a derivative at the corresponding point. This property is useful in solving optimization problems and finding critical points of a function. Additionally, the derivative of the inverse function can be calculated using the chain rule with the derivative of the original function.

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