In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor. For example, a homogeneous real-valued function of two variables x and y is a real-valued function that satisfies the condition for some constant k and all real numbers α. The constant k is called the degree of homogeneity. More generally, if ƒ : V → W is a function between two vector spaces over a field F, and k is an integer, then ƒ is said to be homogeneous of degree k if

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• In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor. For example, a homogeneous real-valued function of two variables x and y is a real-valued function that satisfies the condition for some constant k and all real numbers α. The constant k is called the degree of homogeneity. More generally, if ƒ : V → W is a function between two vector spaces over a field F, and k is an integer, then ƒ is said to be homogeneous of degree k if for all nonzero α ∈ F and v ∈ V. When the vector spaces involved are over the real numbers, a slightly less general form of homogeneity is often used, requiring only that () hold for all α > 0. Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used in the definition of sheaves on projective space in algebraic geometry. More generally, if S ⊂ V is any subset that is invariant under scalar multiplication by elements of the field (a "cone"), then a homogeneous function from S to W can still be defined by (). (en)
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• 979052538 (xsd:integer)
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• Eric Weisstein (en)
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• p/h047670 (en)
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• Suppose that the function is continuously differentiable. Then is positively homogeneous of degree if and only if : (en)
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• Euler's homogeneous function theorem. (en)
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• This result follows at once by differentiating both sides of the equation with respect to , applying the chain rule, and choosing to be . The converse is proved by integrating. Specifically, let . Since , : Thus, . This implies . Therefore, : is positively homogeneous of degree . (en)
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• Homogeneous function (en)
• Euler's Homogeneous Function Theorem (en)
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• EulersHomogeneousFunctionTheorem (en)
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• In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor. For example, a homogeneous real-valued function of two variables x and y is a real-valued function that satisfies the condition for some constant k and all real numbers α. The constant k is called the degree of homogeneity. More generally, if ƒ : V → W is a function between two vector spaces over a field F, and k is an integer, then ƒ is said to be homogeneous of degree k if (en)
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• Homogeneous function (en)
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