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Symbolic vectors and differentiation


Hello world,

Does anybody know how to define or simulate "symbolic vectors",
i.e. something like

	v = Table[i, {i, 1, n}]

but where n is a symbolic constant?  All regular vector and matrix
operations like Dot and Cross should still work, e.g. given above
definition,

	v.v

should yield something like

	Sum[i^2, {i, 1, n}]  or  n/6 (n+1)(2n+1).

A related problem I'm currently facing: Is there a way to make
Mathematica differentiate with respect to indexed variables?  For
example,

	D[Sum[x[i]^2, {i, 1, n}], x[j]]

should yield

	2 x[j]

rather than the actual output 0.

Any comments are highly appreciated.

Ralph


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