# Representing a function in a different space

• Apteronotus
In summary, The conversation discusses changing the reference frame of an implicit function f(x,y,z) in the XYZ Cartesian reference frame to the X'Y'Z' reference frame using a matrix M. It is mentioned that a vector v in XYZ can be represented in the X'Y'Z' reference frame as v' = Mv. The question then arises about how to represent the function f in the X'Y'Z' reference frame. The solution is to use the inverse of the matrix M to transform f as a scalar.
Apteronotus
I have an implicit function f(x,y,z) which represents a surface in the XYZ Cartesian reference frame. I would like to change this current XYZ reference frame by a matrix M.
ie.
$M: XYZ \rightarrow X'Y'Z'$

If I have a vector v in XYZ, then v'=Mv is my representation of v in the X'Y'Z' reference frame. But how do I get a representation of my function f in X'Y'Z'? Specially, as f is given in terms of (x,y,z) and cannot easily be solved for each of its components.

Thanks

f transforms as a scalar, that means (I put r = (x,y,z)):

f(r) = f '(r ')

and since r ' = Mr

$f'(r')=f(M^{-1}r')$

In other words $f'=f\circ M^{-1}$

## 1. What is the purpose of representing a function in a different space?

Representing a function in a different space allows us to better understand and analyze the behavior of the function. It can also provide insights into the relationships between variables and help us make predictions or solve problems.

## 2. What are some common methods for representing a function in a different space?

There are several methods for representing a function in a different space, including transforming the coordinates of the function, using a change of variables, or using a different basis for the function.

## 3. How does changing the space affect the representation of a function?

Changing the space can change the way the function looks and behaves. For example, a linear function may appear curved when represented in a logarithmic space. It can also reveal hidden patterns or relationships that were not apparent in the original space.

## 4. What are some advantages of representing a function in a different space?

One advantage is that it can simplify the function and make it easier to analyze or manipulate. It can also help us see the function in a new light and potentially lead to new insights or applications. Additionally, representing a function in a different space may be necessary for certain calculations or modeling techniques.

## 5. Are there any limitations to representing a function in a different space?

While representing a function in a different space can be useful, it may not always be appropriate or feasible. Some functions may not be able to be transformed or represented in certain spaces. Additionally, the interpretation of the function may change when represented in a different space, so caution should be taken when applying the results to the original space.

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