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五、场论初步

小题​

  1. 【1989-12-3 分】u⃗=xy2i⃗+yezj⃗+xln⁡(1+z2)k⃗\displaystyle \vec{u}=xy^{2}\vec{i}+ye^{z}\vec{j}+x\ln(1+z^{2})\vec{k}在P(1,1,0)\displaystyle P(1,1,0)处div⁡u⃗=\displaystyle \operatorname{div}\vec{u}=

  2. **【1993-12-3 分】**设 u=ln⁡x2+y2+z2\displaystyle u=\ln\sqrt{x^{2}+y^{2}+z^{2}}，则div⁡(grad⁡ u)=\displaystyle \operatorname{div}(\operatorname{grad}\ u)=

  3. 【2001-1-3 分】r=x2+y2+z2\displaystyle r=\sqrt{x^{2}+y^{2}+z^{2}}，则div⁡(grad⁡ r)∣(1,−2,2)=\displaystyle \operatorname{div}(\operatorname{grad}\ r)\big|_{(1,-2,2)}=

  4. 【2016-1-4 分】A⃗=(x+y+z)i⃗+xyj⃗+zk⃗\displaystyle \vec{A}=(x+y+z)\vec{i}+xy\vec{j}+z\vec{k}，求rot⁡A⃗\displaystyle \operatorname{rot}\vec{A}。

  5. 【2018-1-4 分】F⃗=xyi⃗−yzj⃗+zxk⃗\displaystyle \vec{F}=xy\vec{i}-yz\vec{j}+zx\vec{k}，则rot⁡F⃗(1,1,0)=\displaystyle \operatorname{rot}\vec{F}(1,1,0)=