13 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Differential (mathematics) | 3/4 | https://en.wikipedia.org/wiki/Differential_(mathematics) | reference | science, encyclopedia | 2026-05-05T07:23:48.842935+00:00 | kb-cron |
d
f
p
=
∑
j
=
1
n
D
j
f
(
p
)
(
d
x
j
)
p
.
{\displaystyle df_{p}=\sum _{j=1}^{n}D_{j}f(p)\,(dx_{j})_{p}.}
The coefficients
D
j
f
(
p
)
{\displaystyle D_{j}f(p)}
are (by definition) the partial derivatives of
f
{\displaystyle f}
at
p
{\displaystyle p}
with respect to
x
1
,
x
2
,
…
,
x
n
{\displaystyle x_{1},x_{2},\ldots ,x_{n}}
. Hence, if
f
{\displaystyle f}
is differentiable on all of
R
n
{\displaystyle \mathbb {R} ^{n}}
, we can write, more concisely:
d
f
=
∂
f
∂
x
1
d
x
1
+
∂
f
∂
x
2
d
x
2
+
⋯
+
∂
f
∂
x
n
d
x
n
.
{\displaystyle df={\frac {\partial f}{\partial x_{1}}}\,dx_{1}+{\frac {\partial f}{\partial x_{2}}}\,dx_{2}+\cdots +{\frac {\partial f}{\partial x_{n}}}\,dx_{n}.}
In the one-dimensional case this becomes
d
f
=
d
f
d
x
d
x
{\displaystyle df={\frac {df}{dx}}dx}
as before. This idea generalizes straightforwardly to functions from
R
n
{\displaystyle \mathbb {R} ^{n}}
to
R
m
{\displaystyle \mathbb {R} ^{m}}
. Furthermore, it has the decisive advantage over other definitions of the derivative that it is invariant under changes of coordinates. This means that the same idea can be used to define the differential of smooth maps between smooth manifolds. Aside: Note that the existence of all the partial derivatives of
f
(
x
)
{\displaystyle f(x)}
at
x
{\displaystyle x}
is a necessary condition for the existence of a differential at
x
{\displaystyle x}
. However it is not a sufficient condition. For counterexamples, see Gateaux derivative.
==== Differentials as linear maps on a vector space ==== The same procedure works on a vector space with enough additional structure to reasonably talk about continuity. The most concrete case is a Hilbert space, also known as a complete inner product space, where the inner product and its associated norm define a suitable concept of distance. The same procedure works for a Banach space, also known as a complete normed vector space. However, for a more general topological vector space, some of the details are more abstract because there is no concept of distance. For the important case of a finite dimension, any inner product space is a Hilbert space, any normed vector space is a Banach space and any topological vector space is complete. As a result, we can define a coordinate system from an arbitrary basis and use the same technique as for
R
n
{\displaystyle \mathbb {R} ^{n}}
.
=== Differentials as germs of functions === This approach works on any differentiable manifold. If
U and V are open sets containing p
f
:
U
→
R
{\displaystyle f\colon U\to \mathbb {R} }
is continuous
g
:
V
→
R
{\displaystyle g\colon V\to \mathbb {R} }
is continuous then f is equivalent to g at p, denoted
f
∼
p
g
{\displaystyle f\sim _{p}g}
, if and only if there is an open
W
⊆
U
∩
V
{\displaystyle W\subseteq U\cap V}
containing p such that
f
(
x
)
=
g
(
x
)
{\displaystyle f(x)=g(x)}
for every x in W. The germ of f at p, denoted
[
f
]
p
{\displaystyle [f]_{p}}
, is the set of all real continuous functions equivalent to f at p; if f is smooth at p then
[
f
]
p
{\displaystyle [f]_{p}}
is a smooth germ. If
U
1
{\displaystyle U_{1}}
,
U
2
{\displaystyle U_{2}}
V
1
{\displaystyle V_{1}}
and
V
2
{\displaystyle V_{2}}
are open sets containing p
f
1
:
U
1
→
R
{\displaystyle f_{1}\colon U_{1}\to \mathbb {R} }
,
f
2
:
U
2
→
R
{\displaystyle f_{2}\colon U_{2}\to \mathbb {R} }
,
g
1
:
V
1
→
R
{\displaystyle g_{1}\colon V_{1}\to \mathbb {R} }
and
g
2
:
V
2
→
R
{\displaystyle g_{2}\colon V_{2}\to \mathbb {R} }
are smooth functions
f
1
∼
p
g
1
{\displaystyle f_{1}\sim _{p}g_{1}}
f
2
∼
p
g
2
{\displaystyle f_{2}\sim _{p}g_{2}}
r is a real number then
r
∗
f
1
∼
p
r
∗
g
1
{\displaystyle r*f_{1}\sim _{p}r*g_{1}}
f
1
+
f
2
:
U
1
∩
U
2
→
R
∼
p
g
1
+
g
2
:
V
1
∩
V
2
→
R
{\displaystyle f_{1}+f_{2}\colon U_{1}\cap U_{2}\to \mathbb {R} \sim _{p}g_{1}+g_{2}\colon V_{1}\cap V_{2}\to \mathbb {R} }
f
1
∗
f
2
:
U
1
∩
U
2
→
R
∼
p
g
1
∗
g
2
:
V
1
∩
V
2
→
R
{\displaystyle f_{1}*f_{2}\colon U_{1}\cap U_{2}\to \mathbb {R} \sim _{p}g_{1}*g_{2}\colon V_{1}\cap V_{2}\to \mathbb {R} }
This shows that the germs at p form an algebra. Define
I
p
{\displaystyle {\mathcal {I}}_{p}}
to be the set of all smooth germs vanishing at p and
I
p
2
{\displaystyle {\mathcal {I}}_{p}^{2}}
to be the product of ideals
I
p
I
p
{\displaystyle {\mathcal {I}}_{p}{\mathcal {I}}_{p}}
. Then a differential at p (cotangent vector at p) is an element of
I
p
/
I
p
2
{\displaystyle {\mathcal {I}}_{p}/{\mathcal {I}}_{p}^{2}}
. The differential of a smooth function f at p, denoted
d
f
p
{\displaystyle \mathrm {d} f_{p}}
, is
[
f
−
f
(
p
)
]
p
/
I
p
2
{\displaystyle [f-f(p)]_{p}/{\mathcal {I}}_{p}^{2}}
. A similar approach is to define differential equivalence of first order in terms of derivatives in an arbitrary coordinate patch. Then the differential of f at p is the set of all functions differentially equivalent to
f
−
f
(
p
)
{\displaystyle f-f(p)}
at p.