Question

Euler's theorem on homogeneous function in partial differentiation

Answer 1

Homogeneous Functions, Euler's Theorem and Partial Molar Quantities

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Any function f(x) that possesses the characteristic mapping:

x→λxf(x)→λf(x)

(15.1)

is said to be homogeneous, with respect to x, to degree 1. By the same token, if f(x) obeys the mapping:

x→λxf(x)→λkf(x)

(15.2)

then f(x) is homogeneous to degree “k”. In general, a multivariable function f(x1,x2,x3,…) is said to be homogeneous of degree “k” in variables xi(i=1,2,3,…) if for any value of λ ,

f(λx1,λx2,...)=λkf(x1,x2,...)

(15.3)

For example, let us consider the function:

f(x,y)=xx2+y2

(15.4)

How do we find out if this particular function is homogeneous, and if it is, to what degree? We evaluate this function at x=λx and y= λy to obtain:

f(λx,λy)=λxλ2x2+λ2y2=λ−1