Suppose there is a continuously differentiable multivariable scalar function $f$ such that
for $\forall {\bf{p}} \in {R^n}$
$f\left( {{\bf{p}},{\bf{q}}} \right) > 0$ $\forall {\bf{q}} \in {R^n} - \left\{ 0 \right\}$
$f\left( {{\bf{p}},{\bf{q}}} \right)=0$ if $\bf{q}=\bf{0}$
Namely, $f$ is positive definite with respect to $\bf{q}$.
What kind of sufficient or N-S condition is needed to say that
$\frac{{\partial f}}{{\partial {\bf{q}}}}{\bf{q}} > 0$ $\forall {\bf{p}} \in {R^n}$ $\forall {\bf{q}} \in {R^n} - \left\{ 0 \right\}$
? I think $\frac{{\partial f}}{{\partial {\bf{q}}}} \ne {\bf{0}}$ is only a condition needed. But I am not sure.
You can just ignore $\bf{p}$ in the statement. That is, $f=f\left(\bf{q}\right)$.
Thank you for your help!