Exercises
[1G6]Prerequisites:[09N].Let \(f:ℝ^ k→ℝ\) be of class \(C^∞\). Recall that, by Schwarz’s theorem, permutiation of the order of partial derivatives does not change the result. Let \(N(n,k)\) be the number of partial (potentially different) derivatives of order \(n\): show that \(N(n,k)=\binom {n+k-1}{k-1}\) (which is a polynomial with integer coefficients in the variable \(n\), of order \(k-1\)).
Solution 1