考研数学笔记 —— 多元函数微分学关系图
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考研数学笔记 —— 多元函数微分学关系图

in 高数 with 0 comment, Views is 131

多元函数微分学关系图

【注】

$$ 连续可偏导 \; => \; 可微 \; => \; \begin{cases} => 连续 \\ \\ => 可偏导 \end{cases} $$

$$ z = f_{\left( x, y \right)} = \sqrt{x^2 + y^2} $$

$$ f_x^\prime \left( 0, 0\right) = \lim_{x \rightarrow 0} \dfrac{f_{\left( x, 0 \right)} - f_{\left( 0, 0 \right)}}{x} = \lim_{x \rightarrow 0} \dfrac{\left| x \right|}{x} = \nexists $$

$$ f_y^\prime \left( 0, 0\right) = \lim_{y \rightarrow 0} \dfrac{f_{\left( 0, y \right)} - f_{\left( 0, 0 \right)}}{x} = \lim_{y \rightarrow 0} \dfrac{\left| y \right|}{y} = \nexists $$

$$ z = f_{\left( x, y \right)} = \begin{cases} \; \dfrac{xy}{x^2 + y^2} & , \; f_{\left( x, y\right)} \neq f_{\left( 0, 0 \right)} \\ \\ \; 0 & , \; f_{\left( x, y \right)} = f_{\left( 0, 0\right)} \end{cases} $$

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