document: Differential calculus for p-norms of complex-valued vector functions with applications For complex-valued n-dimensional vector functions t to s -LRB- t -RRB- , supposed to be sufficiently smooth , the differentiability properties of the mapping t to | | s -LRB- t -RRB- | | / sub p / at every point t = t/sub 0 / epsilon R/sub 0 / / sup + / : = -LCB- t epsilon R | t > or = 0 -RCB- are investigated , where | | . | | / sub p / is the usual vector norm in C/sup n / resp . R/sup n / , for p epsilon -LSB- 1 , o infinity -RSB- . Moreover , formulae for the first three right derivatives D/sub + / / sup k / | | s -LRB- t -RRB- | | / sub p / , k = 1 , 2,3 are determined . These formulae are applied to vibration problems by computing the best upper bounds on | | s -LRB- t -RRB- | | / sub p / in certain classes of bounds . These results can not be obtained by the methods used so far . The systematic use of the differential calculus for vector norms , as done here for the first time , could lead to major advances also in other branches of mathematics and other sciences
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