I'm trying to see how two ideas are related:
In (Discrete) Exterior Calculus, we defined k-vectors as volumes, and that k-forms are like "measurement tools", that allow us to measure k-vectors.
However, in Differential geometry, for a real differentiable $n$-manifold $M$, and $p \in M$, we first defined the tangent space $T_pM$, using either tangent-curves or derivations. We've proved that for a chart $(U,x)$, where $U$ is an open set of $M$, a basis for $T_p M$ is: $$ \{ \frac{\partial}{\partial x^i} \} \quad , i = 1 \ldots, n$$ Then we've defined the $k$th exterior power $\bigwedge ^k (T_p M)$, and we called its elements $(k,0)$-tensors.
What is the relation between the $(k,0)$-tensors of differential geometry and the $k$-vectors defined in exterior algebra? Can we say that wedging partial-differential operators are like "volumes" then?