Of course one could express the Taylor series of a vector-valued function as a collection of Taylor-series of its scalar-valued components. But this seems awkward to me and is not a very compact notation.

The linear term of the Taylor expansion of a vector-field can be expressed concisely in terms of the Jacobian. So I wanted to know if something similar can be done for the quadratic term.

You mean something like F(x+h)=F(x)+F'(x)h+1/2*F''(x){h,h} + 1/6*F'''(x){h,h,h} + ...?

The type of parantheses may vary, but it is good practice to use different types for the nonlinear argument, i.e., the evaluation point, and the (multi-)linear arguments.

Perhaps this link will help you, look last equation:

https://people.fh-landshut.de/~maurer/numeth/node44.html

It is always helpful to use matrix notation.

Thanks everyone!

@Nadeem: how would such a tensor object look like (is it just arranging the Hessians in a tensor (--> third order))? And how would the vectors of the variables then multiply with this tensor?

Until Nadeem comes up with a better variant:

H^i_{\ jk} u^jv^k

with upper and lower indices, not powers. u^i is the contravariant coordinate vector of a tangent vector, a_j would represent the covariant coordinate vector of a linear form.

Have a look at here:

https://www.physicsforums.com/threads/taylor-expansion-of-a-vector-field.626350/

It has been quite long since the original question was posed, but if anybody is still interested, I think the answer is in the first formula of Section A.1.2 in this paper

http://www3.stat.sinica.edu.tw/sstest/oldpdf/A21n215.pdf