Why does $|(d\exp_{p})_{v}(w)|$ encodes the rate of spreading of geodesics? I'm reading a discussion in Do Carmo's "Riemannian Geometry", about Jacobi's equation.
The writer uses the following notation:
$$v(s):I\to T_{p}M\ \ \ s.t. \\
v(0)=v,\\
v'(0)=w
$$ We identify $T_pM\simeq T_vT_p M $.
The writer motivates the discussion by the following claim:

We would like to obtain information on $|(d\exp_{p})_{v}(w)|$. One of the reasons for this is that $|(d\exp_{p})_{v}(w)|$ denotes, intuitively, the rate of spreading of the geodesics:
$$ t \to \exp_{p}(t v(s)) 
$$

I don't understand the heuristic above. To be concrete, I'd like to know:

*

*The spread of which geodesics is encoded in $|(d\exp_{p})_{v}(w)|$? Is it some property that relates the following family of geodesics: $$\{t \to \exp_{p}(t v(s)) \}_{s\in I}$$

*I'd be glad to hear a bit more about that - Why exactly does the differential encodes spread of geodesics? What is the geometric interpretation of all that?

*Could anyone point me to some draw of figure explaining this heuristic graphically?

It all sort of make sense, but I think some more details could be of tremendous help.
Many Thanks!
 A: Disclaimer: This answer is an attempt to explain the intuition behind Do Carmo's statement.
There will be a lot of abuses of notations, which could be justified on a vector space and therefore, in local coordinates.
For instance, we will perform additions on a manifold and differences between two points whose results are tangent vectors.
I believe too rigorous notations would have resulted in a loss of clarity.
This stunt is performed by a professional.
Don't do this at home.

Fix $p\in M$, and consider two vectors $v,v' \in T_pM$.
Let $w=v'-v$.
If $w$ is small (that is, if $v$ an $v'$ are close), then intuitively
$$
\exp_p(v')=\exp_p(v+w) \simeq \exp_p(v) + (d\exp_p)_vw,
$$
(this is the meaning of the derivative of a function) from which one would like to write,
$$
\exp_p(v') - \exp_p(v) \simeq (d\exp_p)_vw.
$$
With a total violation of the concept of rigour, let us write
$$
|\exp_p(v') - \exp_p(v)| \simeq |(d\exp_p)_vw|.
$$
In some more rigorous notations (but that we haven't justified), this means
$$
d_g\left(\exp_p(v+w),\exp_p(v)\right) \simeq \|(d\exp_p)_v(w)\|
$$
when $w$ is small.
See the picture below, where $\gamma_{p,v}(t) = \exp_p(tv)$ and $\gamma_{p,v'}(t) = \exp_p(tv')$.

If $v(s)$ is a family of tangent vectors in $T_pM$, define $\gamma_s(t) = \exp_p(tv(s))$.
Then $\{\gamma_s\}$ is a family of geodesics passing through $p$ and whose initial data is $\gamma'_s(0)=v(s)$.
See the picture below.

From the definition of the derivative of a function, one would like to write
$$
\gamma_s(t) \simeq \gamma_0(t) + s\left.\frac{\partial \gamma_s(t)}{\partial s}\right|_{s=0},
$$
which, from the discussion above, brings us, for small $s$,
$$
d_g\left(\gamma_s(t),\gamma_0(t)\right) \simeq |s| \left\|\left.\frac{\partial \gamma_s(t)}{\partial s}\right|_{s=0}\right\|.
$$
Hence, $\left.\frac{\partial \gamma_s(t)}{\partial s}\right|_{s=0}$ encodes how distant are the points $\{\gamma_s(t)\}$, for $t$ fixed and $s$ moving around $0$.
In other word, the vector field $t\mapsto \left.\frac{\partial \gamma_s(t)}{\partial s}\right|_{s=0}$ along the curve $\gamma_0$ encodes how the different curves $\{\gamma_s\}$ are distant from $\gamma_0$ at first order.
From an easy calculation, we have
$$
\left.\frac{\partial \gamma_s(t)}{\partial s}\right|_{s=0} = \left(d\exp_p\right)_{tv(0)}(tv'(0)).
$$
It follows that, on an intuitive level, $\left(d\exp_p\right)_{tv(0)}(tv'(0))$ encodes how distant are the point $\gamma_s(t)$ from $\gamma_0(t)$, for $s$ small.
Do Carmo is considering this at time $t=1$, and
the bigger $\left\|\left(d\exp_p\right)_{v(0)}(v'(0))\right\|$ is, the bigger is the distance between $\exp_p(v)$ and $\exp_p(v(s))$ for small $s$.
The relevant, rigorous and useful concept for studying the rate of spreading of a family of geodesics is the concept of Jacobi fields.
I suspect that Do Carmo introduces them just after this discussion.
