Yes, you can. However in spacial cases. For instance if f(x,y) obeys scaling low.

In priciples you can assume that it does and then you have to make an experimental test. f(x,y) obeys the scaling low if this is homogenous function in general sense, which means that there exists real a,b,c such that for each lambda>0 the following relation holds: f(lambdaa x,lambdab y)=lambdac f(x,y). According to assumptions we are free to substitute: lambda=x-1/a. Then we obtain: 

f(1,x^(-b/a)y)=x^(-c/a)f(x,y). On the left hand side you have got function of the one variable z=x^(-b/a)y.

Thank you so much...this is what I was looking for..

What is the name of this theory and is there a reference book? Because I want a more generalized case i.e f(x1,...,xn) converted to f(y1,...yp) such that p

In some cases you can use the polar coordinates, where r=sqrt(x^2+y^2), but you need also a kind of symmetry in order to get rid of polar angle theta...

Name of theory is SCALING. By the scaling one understands an invariance of analytic properties with respwct to transformation which I have described in my previous message. Book by G.I Barenblat : "SCALING", Cambridge Texts in Applied Mathematics., Cambridge University Press. I suggest also book by E. Stanley, "Phase transitions and Critical Phenomena. Do the best.

It's possible that I haven't understood exactly what you want to accomplish, but it is clear that in general the answer to your question is "no". If it were possible to reduce the number of variables always, by some method, then there would hardly be any need to study functions of more than one variable.

Yes, you can repeat the scaling algoritm for the multi variables function:

f(lambda^a x, lambda^b y, lambda^c z.....)=lambda^k f(x,y,z,.....). Let us drop out the second argument: lambda=y^(-1/b), -> f(y^(-a/b) x, 1, y^(-c/b) z, ...)=y^(-k/b) f(x,y,z,...).

et cetera.

You always can  but it is not free and you cannot do it keeping invariance under addition: There is a surjective continuous map

H:[0,1] \to [0,1] \times [0,1]

the so called Hilbert plane filling curve. That map is not differentiable and it is not one to one.  Therefore a function f(x,y) on [0,1] \times [0,1] is completely determined by (f\circ H)(t) on [0,1].

However if you want to replace a differentiable function with a differentiable function of one variable less, you really do need some  symmetry, like homogeneity (f effectively depends only on the angle with the x axis)  or rotational invariance (f effectively depends only on the radius).   

Of course, one can always "reduce" a function of several arguments to a function of a single argument, since product and Hom are adjoint functors, that is, since there is a natural transformation between

hom(A x B, C)

and

hom(A, C^B)

This general property was known to Frege and Schoenfinkel. My answer is really a response to the statement made by prior to mine, regarding the statement

"If it were possible to reduce the number of variables always, by some method, then there would hardly be any need to study functions of more than one variable."

I think in some cases, it is possible but if we could do this in general case, the theory of n variable functions was redundant. As I know, for some special positive n-linear map on C(K) spaces where K is a compact Hausdorff space, there is a way to reduce the variable. Please see my joint paper with Vladimir Troitsky : doi: 10.1007_s11117-013-0239-3, Theorem 1.

I hope this would be useful.

Thank you all

Rogier Brussee  and Mycroft Holmes , can you please suggest some text books or papers that talks about these theories when used to reduce the number of variables of a function? I would like to read in depth about these theories.

Thank you

The Wikipedia articles  https://en.wikipedia.org/w/index.php?title=Hilbert_curve&oldid=633637210 and https://en.wikipedia.org/w/index.php?title=Space-filling_curve&oldid=624328726 are actually quite good for a practical  and theoretical overview.

Mycroft Holmes trick is called adjunction in mathematics and currying in computer science. It is a very simple trick: 

given f: X \times Y \to Z

we have  for every x \in X a function  

F_x : Y \to Z ,

F_x: y  \to f(x,y)

and so we have map

\phi_f : X \to Map(Y,Z), 

x \to F_x 

Note that unlike the Hilbertcurve construction this construction is natural in the sense that it is unique and behaves well with respect to maps X\times Y \to X' \times Y'

This is the version for the category of sets. Things get more interesting if we work in different categories, because we now have to define "products"  and "Hom objects" in the category. (in most categories maps are called morphisms or homomorphisms) . Two examples:

In finite dimensional vectorspaces we have a "product" of two vectorspaces, the tensor product V\tensor W[1] and Hom object Hom(V,W) is just the vectorspace of  linear maps from V  to W. We have the adjunction map:

LInmaps(U \tensor V, W) \iso Linmaps( U, Hom(V,W))

In the category of topological spaces we have a product X \times Y of two topological spaces  X and Y. We also have the space of continuous maps Map(X,Y) where the topology is defined by defining an open set as unions of finite intersections of open sets V(K,U) consisting of all maps that map a compact set K in an open set U. For locally compact spaces X,Y,Z we then have

ContinuousMaps(X \times Y, Z) \iso ContinuousMaps(X, Map(Y,Z))

Thank you so much

Would this work for multi variable function f(x, y, z,......)  not only XYZ function? 

As Far  as I understand this Map is only defined for 3D space.  

Well you can do

Map(X_1 \times X_2 \times X_3 , Y)

           \iso Map(X_1, Map(X_2\times X_3, Y))

            \iso Map(X_1, Map( X_2, Map(X_3, Y)))

I think in some functional programming languages (Haskell IIRC) they even do this in practice, so  they write the "Curried" form

f: int -> string -> bool -> double

which means f: int -> (string -> (bool -> double)))

for what in most computer languages you would write as 

f(int, string, bool) -> double 

or C/Java  style

double f(int, string, bool)