I was unsure whether to ask this here or at a physics SE.
Wald's "General Relativity" defines parallel transport as follows:
$\nabla$ is a derivative operator (is linear, obeys Leibniz rule, commutative with contraction, torsion free and is consistent with the notion of vectors as directional derivatives). A vector $v^b$ given at each point of a curve C is parallel transported is said to be parallelly transported as one moves along the curve if the equation
$t^a \nabla _a v^b =0$
is satisfied along the curve, where $t^a$ are vectors tangent to the curve.
How does this definition reproduce what we intuitively understand as parallel transport? Also, other references use different terminology, with a $\nabla _v$ meaning the derivative along a vector $v$, (the same role $t^a$ plays in the definition above) which is easier to understand but (1) seems ill defined, unlike wald's definition, and (2) both definitions should be related in some way.
Best Answer
When a thing is parallel transported, doesn't it remain in a sense unchanged?
Intuitively the equation $$ t^a \nabla _a v^b =0 $$ means that vector $v$ is constant (the derivative is zero!) with respect to differentiation along the velocity vector of curve $C$. This velocity vector $$ t=\dot{C} $$ provides a natural way of expressing the direction in which vector $v$ is being transported.
Which vector $v$ is meant in your last paragraph? It looks to me that there is a confusion. Wald's definition can be written in terms of derivatives along vectors as well, just slightly different notation is used. Namely, $$ \nabla_{t} v^b:= t^a \nabla_ a v^b $$ Notice that Wald uses so called abstract index notation, while other sources that you assume may use "invariant" or "coordinate" ways of writing down that same things.