最新微分几何(第三版)梅向明黄敬之编[1]
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微分几何(第三版)梅向明黄敬之编[1]
第一章曲线论
§2 向量函数
5. 向量函数«Skip Record If...»具有固定方向的充要条件是«Skip Record If...»× «Skip Record If...»= «Skip Record If...»。
分析：一个向量函数«Skip Record If...»一般可以写成«Skip Record If...»=«Skip Record
If...»«Skip Record If...»的形式，其中«Skip Record If...»为单位向量函数，«Skip Record If...»为数量函数，那么«Skip Record If...»具有固定方向的充要条件是«Skip Record If...»具有固定方向，即
«Skip Record If...»为常向量，（因为«Skip Record If...»的长度固定）。
证对于向量函数«Skip Record If...»，设«Skip Record If...»为其单位向量，则«Skip Record
If...»=«Skip Record If...»«Skip Record If...»，若«Skip Record If...»具有固定方向，则«Skip Record If...»为常向量，那么«Skip Record If...»=«Skip Record If...»«Skip Record If...»，所以«Skip Record If...»×«Skip Record If...»=«Skip Record If...»«Skip Record If...»（«Skip Record If...»×«Skip Record If...»）
=«Skip Record If...»。
反之，若«Skip Record If...»×«Skip Record If...»=«Skip Record If...»，对«Skip Record If...»=«Skip Record If...»«Skip Record If...»求微商得«Skip Record If...»=«Skip Record If...»«Skip Record
If...»+«Skip Record If...»«Skip Record If...»，于是«Skip Record If...»×«Skip Record If...»=«Skip Record If...»（«Skip Record If...»×«Skip Record If...»）=«Skip Record If...»，则有 «Skip Record If...» = 0 或«Skip Record If...»×«Skip Record If...»=«Skip Record If...»。当«Skip Record If...»= 0时，«Skip Record If...»=«Skip Record If...»可与任意方向平行；当«Skip Record If...»«Skip Record If...»0时，有«Skip Record If...»×«Skip Record If...»=«Skip Record If...»，而（«Skip Record If...»×«Skip Record If...»«Skip Record If...»=«Skip Record If...»-(«Skip Record If...»·«Skip Record If...»«Skip Record If...»＝«Skip Record If...»，（因为«Skip Record If...»
具有固定长， «Skip Record If...»·«Skip Record If...»= 0），所以 «Skip Record If...»=«Skip Record If...»，即«Skip Record If...»为常向量。所以，«Skip Record If...»具有固定方向。
6．向量函数«Skip Record If...»平行于固定平面的充要条件是（«Skip Record If...»«Skip Record If...»«Skip Record If...»）=0 。
分析：向量函数«Skip Record If...»平行于固定平面的充要条件是存在一个定向向量«Skip Record If...»，使«Skip Record If...»·«Skip Record If...» = 0 ，所以我们要寻求这个向量«Skip Record If...»及«Skip Record If...»与«Skip Record If...»，«Skip Record If...»的关系。
证若«Skip Record If...»平行于一固定平面π，设«Skip Record If...»是平面π的一个单位法向量，则
«Skip Record If...»为常向量，且«Skip Record If...»·«Skip Record If...» = 0 。两次求微商得«Skip Record If...»·«Skip Record If...» = 0 ，«Skip Record If...»·«Skip Record If...» = 0 ，即向量«Skip Record If...»，«Skip Record If...»，«Skip Record If...»垂直于同一非零向量«Skip Record If...»，因而共面，即（«Skip Record If...»«Skip Record If...»«Skip Record If...»）=0 。
反之, 若（«Skip Record If...»«Skip Record If...»«Skip Record If...»）=0，则有«Skip Record If...»×«Skip Record If...»=«Skip Record If...»或«Skip Record If...»×«Skip Record If...»«Skip Record If...»«Skip Record If...»。若«Skip Record If...»×«Skip Record If...»=«Skip Record If...»，由上题知«Skip Record If...»具有固定方向，自然平行于一固定平面，若«Skip Record If...»×«Skip Record If...»«Skip Record If...»«Skip Record If...»，则存在数量函数«Skip Record If...»、«Skip Record If...»，使«Skip Record If...»= «Skip Record If...»+«Skip Record If...»«Skip Record If...»①
令«Skip Record If...»=«Skip Record If...»×«Skip Record If...»，则«Skip Record If...»«Skip Record If...»«Skip Record If...»，且«Skip Record If...»⊥«Skip Record If...»。对«Skip Record If...»=«Skip Record If...»×«Skip Record If...»求微商并将①式代入得«Skip Record If...»=«Skip Record If...»×«Skip Record If...»=«Skip Record If...»（«Skip Record If...»×«Skip Record If...»）=«Skip Record If...»«Skip Record If...»，于是«Skip