In the math branches of differential geometry and vector calculus, the **second covariant derivative**, or the **second order covariant derivative**, of a vector field is the derivative of its derivative with respect to another two tangent vector fields. Formally, given a (pseudo)-Riemannian manifold (*M*, *g*) associated with a vector bundle *E* → *M*, let ∇ denote the Levi-Civita connection given by the metric *g*, and denote by Γ(*E*) the space of the smooth sections of the total space *E*. Denote by *T*^{*}M the cotangent bundle of *M*. Then the second covariant derivative can be defined as the composition of the two ∇s as follows:

Γ
(
E
)
⟶
∇
Γ
(
T
∗
M
⊗
E
)
⟶
∇
Γ
(
T
∗
M
⊗
T
∗
M
⊗
E
)
.
For example, given vector fields *u*, *v*, *w*, a second covariant derivative can be written as

(
∇
u
,
v
2
w
)
a
=
u
c
v
b
∇
c
∇
b
w
a
by using abstract index notation. It is also straightforward to verify that

(
∇
u
∇
v
w
)
a
=
u
c
∇
c
v
b
∇
b
w
a
=
u
c
v
b
∇
c
∇
b
w
a
+
(
u
c
∇
c
v
b
)
∇
b
w
a
=
(
∇
u
,
v
2
w
)
a
+
(
∇
∇
u
v
w
)
a
.
Thus

∇
u
,
v
2
w
=
∇
u
∇
v
w
−
∇
∇
u
v
w
.
One may use this fact to write Riemann curvature tensor as follows:

R
(
u
,
v
)
w
=
∇
u
,
v
2
w
−
∇
v
,
u
2
w
.
Similarly, one may also obtain the second covariant derivative of a function *f* as

∇
u
,
v
2
f
=
u
c
v
b
∇
c
∇
b
f
=
∇
u
∇
v
f
−
∇
∇
u
v
f
.
Since Levi-Civita connection is torsion-free, for any vector fields *u* and *v*, we have

∇
u
v
−
∇
v
u
=
[
u
,
v
]
.
By feeding the function *f* on both sides of the above equation, we have

(
∇
u
v
−
∇
v
u
)
(
f
)
=
[
u
,
v
]
(
f
)
=
u
(
v
(
f
)
)
−
v
(
u
(
f
)
)
.
∇
∇
u
v
f
−
∇
∇
v
u
f
=
∇
u
∇
v
f
−
∇
v
∇
u
f
.
Thus

∇
u
,
v
2
f
=
∇
v
,
u
2
f
.
That is, the value of the second covariant derivative of a function is independent on the order of taking derivatives.