implicit function f(x,y) and z=f(x,y)

Hi

In my mind I'm confusing an implicit function f(x,y) with function z=f(x,y). One of the characteristics of an implicit is that it can let us represent a two-part function in one expression. For instance, consider the equation of a circle: x^2 + y^2 = c^2 [one part function] => y=+/- sqrt(c^2-x^2) [two part function]. Are there some other advantages of an implicit function? In case of an implicit function, y is a dependent variable (function of x) and x is independent.

Isn't function in space also represented as z=f(x,y) where 'z' is determined using x and y coordinates? Is y an independent variable or function of x as it is for an implicit function in x-y plane? Perhaps, both x and y are functions of some other parameter such as 'c'.

I'm sorry if my query(ies) is too confusing. Please let me know if you need some clarification about any part. Please help me with it. Thank you.

Regards
PG

Hi,

An implicit function in x and y is two dimensional, while z=f(x,y) is three dimensional.
In the two dimensional case, x and y are in one plane and that's it. In the three dim case the function produces a third dimension quantity that depends on both x and y.

Examples:
r^2=x^2+y^2 (this is implicit in x and y, R is a constant, more properly written R^2=x^2+y^2)

z=x^2+y^2 (this is three dimensional, z is a variable).

Implicit differentiation:
y^2=2y+x+2
2yy'=2y'+1

Differentiation on z=f(x,y):
z=x^2+y^3
dz/dx=2*x
dy/dx=3*y^2

Implicit functions allow the expression of multivalued functions without declaring two or more different functions.
For example
R^2=x^2+y^2
allows us to define an entire circle in one expression rather than having to define it in two expressions:
y=sqrt(R^2-x^2) (top half)
y=-sqrt(R^2-x^2) (bottom half)