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? asked in 科學數學 · 1 decade ago

functions of several variables

if h is a function of x and y, then there are functions f and g of one variable such that

h(x,y) = f(x) + g(y)

if f is a function of x and y and a is a real number, then

f(ax,ay) = af(x,y)

這些敘述對嗎

錯在哪邊呢?

2 Answers

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  • 1 decade ago
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    Cosider h(x,y) = xy.

    If f(x) + g(y) = h(x,y) = xy,

    f(x) + g(0) = 0 and f(0) + g(y) = 0,

    f(x) = -g(0) and g(y) = -f(0),

    f(x) + g(y) = constant, a contradiction.

    Thus in general it is false.

    Cosider f(x,y) = x + 1.

    If f(ax,ay) = af(x,y),

    ax + 1 = a(x + 1)

    a = 1

    Thus in general it is false.

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  • 1 decade ago

    這些敘述皆錯.反例到處都是,比方說 f(x,y)=x^2*y.

    第二種情形成立的話有個名稱:f(x,y)是個齊次函數,次數是一(homogeneous function of degree1). 我的反例是個次數為3的.

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