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In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; the power is called the degree of homogeneity, or simply the degree.
A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form.
A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if φ(x) is a solution, so is cφ(x), for any (non-zero) constant c. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or ...
A complete classification of homogeneous distributions in one dimension is possible. The homogeneous distributions on R \ {0} are given by various power functions.In addition to the power functions, homogeneous distributions on R include the Dirac delta function and its derivatives.
where , , are respectively homogeneous functions of degree 0, 1, and 2 in generalized velocities. If this system is scleronomous, then the position does not depend explicitly with time: If this system is scleronomous, then the position does not depend explicitly with time:
By Euler's second theorem for homogeneous functions, ¯ is a homogeneous function of degree 0 (i.e., ¯ is an intensive property) which ...
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(This is equivalent to saying that intensive composite properties are homogeneous functions of degree 0 with respect to {}.) It follows, for example, that the ratio of two extensive properties is an intensive property.