**1) The vector derivative, $\partial$**

In geometric calculus, one deals in not just vector fields but multivector fields--fields that associated oriented planes, volumes, or other types of primitives to each point. These multivector fields are differentiated by an operator denoted $\partial$. It can act on multivector fields in either of two ways. On a multivector field $A(r)$, it can act as $\partial \wedge A$, which is the familiar exterior derivative. This increases the grades of all components of the field by one--vectors become planes, planes become volumes, and so on.

But there is another derivative, denoted $\partial \cdot A$, which goes by various names: interior derivative, codifferential, and so on. Both these notions of differentiation arise from $\partial$, however. It is, in my opinion, foolish that differential forms treats the $\partial \cdot$ operation as somehow only expressible in terms of $\partial \wedge$, however. To me (and in GC) they are on equal footing with one another.

**2) The covariant derivative, $\nabla$**

Now, introduce a global rotation field called $\underline R(a; r)$, which acts linearly on the vector $a$ and is a function of position $r$. For brevity, we'll just call this $\underline R(a)$ in most cases. We can, at our discretion, use or set this rotation field to our liking, perhaps because it is convenient, perhaps because it is necessary--you can regard it as inherent to the space if you like.

We can then look at the transformation of $A \mapsto A' = \underline R(A)$. This naturally changes the way we must differentiate. See that

$$a \cdot \partial A' = \underline R(a \cdot \partial A) + (a \cdot \dot \partial) \dot{\underline{R}}(A)$$

This is just a fancy product rule, with the overdot saying we differentiate only the linear operator, not its argument.

We **define** the covariant derivative to get rid of the messy second term on the right-hand side. That is,

$$a \cdot \nabla A' = \underline R(a \cdot \nabla A)$$

Introducing or changing the rotation field changes the covariant derivative. This gives a way to talk about differentiation regardless of the current rotation field $\underline R$. Changing the rotation field can be beneficial to alter the geometry of the space in a way that is convenient. Thus, the rotation field represents generalized, position-dependent rotational degrees of freedom to rotate fields at all points in space by varying amounts and orientations at will. The covariant derivative allows us to do this and still recover results that are independent of the choice of rotation field--of the choice of gauge.

**3) The Lie derivative**

In GC, the Lie derivative has no special symbol. Rather, it can be built from covariant derivatives. Consider two vector fields $A, B$. The Lie derivative is simply

$$\mathcal L_A B = A \cdot \nabla B - B \cdot \nabla A$$

I'm not as familiar with Lie derivatives, but I'm given to understand that if $B$ were transported along a "flow" generated by $A$, this quantity would measure how much $B$ maintains its value during the process.