一、题目
求解下面这个函数的全微分:
$$
z = \frac{y}{\sqrt{x^{2} + y^{2}}}
$$
难度评级:
二、解析
已知:
$$
z = \frac{y}{\sqrt{x^{2} + y^{2}}} = \frac{y}{(x^{2} + y^{2})^{\frac{1}{2}}} \Rightarrow
$$
$$
z = y \cdot (x^{2} + y^{2})^{-\frac{1}{2}}
$$
Next
则:
$$
\frac{\partial z}{\partial x} = -\frac{1}{2} y \cdot 2x (x^{2} + y^{2})^{-\frac{3}{2}} \Rightarrow
$$
$$
\frac{\partial z}{\partial x} = \frac{-xy}{(x^{2} + y^{2})^{\frac{3}{2}}}
$$
Next
继续:
$$
\frac{\partial z}{\partial y} = \frac{\sqrt{x^{2} + y^{2}} – y \cdot \frac{1}{2} \cdot 2y (x^{2} + y^{2})^{-\frac{1}{2}}}{x^{2} + y^{2}} \Rightarrow
$$
$$
\frac{\partial z}{\partial y} = \frac{\sqrt{x^{2} + y^{2}} – \frac{y^{2}}{\sqrt{x^{2} + y^{2}}}}{x^{2} + y^{2}} \Rightarrow
$$
$$
\frac{\partial z}{\partial y} = \frac{\sqrt{x^{2} + y^{2}} \big(\sqrt{x^{2} + y^{2}} – \frac{y^{2}}{\sqrt{x^{2} + y^{2}}} \big)}{\sqrt{x^{2} + y^{2}}(x^{2} + y^{2})} \Rightarrow
$$
$$
\frac{\partial z}{\partial y} = \frac{x^{2} + y^{2} – y^{2}}{(x^{2} + y^{2})^{\frac{3}{2}}} \Rightarrow
$$
$$
\frac{\partial z}{\partial y} = \frac{x^{2}}{(x^{2} + y^{2})^{\frac{3}{2}}}
$$
Next
于是:
$$
\mathrm{d} z = \frac{\partial z}{\partial x} \mathrm{d} x + \frac{\partial z}{\partial y} \mathrm{d} y \Rightarrow
$$
$$
\mathrm{d} z = \frac{-xy}{(x^{2} + y^{2})^{\frac{3}{2}}} \mathrm{d} x + \frac{x^{2}}{(x^{2} + y^{2})^{\frac{3}{2}}} \mathrm{d} y \Rightarrow
$$
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