grad f(1? 1? 1)?6i?3j?

9? 设u? v都是 x? y? z的函数? u? v的各偏导数都存在且连续? 证明

(1) grad(u?v)?grad u? grad v? 解 grad(u?v)??(u?v)?(u?v)?(u?v)i?j?k ?x?y?z?x?x?y?y?z?z ?(?u??v)i?(?u??v)j?(?u??v)k ?(?ui??uj??uk)?(?vi??vj??vk) ?x?y?z?x?y?z ?gradu?gradv? (2) grad (uv)?vgrad u?ugrad v? 解 grad(uv)??(uv)?(uv)?(uv)i?j?k ?x?y?z?x?x?y?y?z?z ?(v?u?u?v)i?(v?u?u?v)j?(v?u?u?v)k ?v(?ui??uj??uk)?u(?vi??vj??vk) ?x?y?z?x?y?z ?vgrad u ?ugrad v? (3) grad (u2)?2ugrad u?

?u2?u2?u2 解 grad(u)?i?j?k ?x?y?z2 ?2u?ui?2u?uj?2u?uk ?x?y?z ?2u(?ui??uj??uk)?2ugradu? ?x?y?z

10? 问函数u?xy2z在点p(1? ?1? 2)处沿什么方向的方向导数最大? 并求此方向导数的最大值?

解 grad u??ui??uj??uk?y2zi?2xyzj?xy2k? ?x?y?z grad u(1, ?1, 2)?(y2zi?2xyzj?xy2k)(1,?1,2)?2i?4j?k? grad u(1? ?1? 2)为方向导数最大的方向? 最大方向导数为 |grad u(1, ?1, 2)|?22?(?4)2?12?21?