In geometric calculus I see that we can unify the three fundamental derivatives from vector calculus; the gradient, the curl, and the diverge; into one operator. However, $\nabla$ is defined for any multivector field, not just scalar and vector fields. What does the geometric calculus gradient of a multivector field really mean? For instance, if $B$ is a bivector field on an open set $D\subseteq \Bbb R^n$ then what does $\nabla B$ tell us? It should be the case that $\nabla B = \nabla \cdot B + \nabla \wedge B$, but then that just leads to the question: what do the divergence and curl of $k$-vector fields, where $k\gt 1$, mean?
As an example of the type of info I'm looking for, just consider the geometric meaning the usual vector calculus derivatives:
- The gradient of a scalar field is the direction of steepest ascent.
- The divergence of a vector field is flux per unit volume through an sufficiently small volume around a point
- The curl of a vector field is the circulation per unit area in a sufficiently small area of a plane around a point (alternatively: the circulation around the normal vector in $\Bbb R^3$)
With these intuitive concepts at hand I know roughly what I get from performing any of these operations. But when it comes to geometric calculus, I know how to take the gradient of a bi(/$k$-)vector field but I don't know what information it should be telling me.