Geoscience Reference
In-Depth Information
CALCULUS
lim
δ
→
0
f(x
+
δ)
−
f(x)
df
is
f
(x)
The
derivative
(slope
of
tangent
line)
of
f
at
x
=
dx
=
.
δ
d
2
f
dx
2
df
dx
. For
at
x
is
f
(x)
The
second
derivative
(curvature)
of
f
=
=
any
constant
c
,
c
=
0,
(cx)
=
c
,
(x
a
)
=
ax
a
−
1
,
(
log
x)
=
1
/x
,
(e
x
)
=
g(x))
=
f
(x)
g
(x)
,
e
x
,
(f (x)
+
+
(f (x)g(x))
=
f
(x)g(x)
+
f(x)g
(x)
.
df (y)
dx
df (y)
dy
dy
The chain rule:
=
dx
.
∂f
∂x
i
The
partial
derivative
of
multivariate
function
f(x
1
,...,x
n
)
w.r.t.
x
i
is
=
∂x
n
.
The gradient is a vector in the same space as
x
. It points to a “higher ground” in terms of
f
value.
∂f
lim
δ
→
0
f(x
1
...x
i
+
δ...x
n
)
−
f(x
1
...x
i
...x
n
)
δ
∂x
1
...
∂f
=
(x
1
,...,x
n
)
is
. The gradient at
x
∇
f(
x
)
=
The
second
derivatives
of
a
multivariate
function
form
an
n
×
n
Hessian
matrix
∂x
i
∂x
j
i,j
=
1
...n
.
∂
2
f
2
f(
x
)
=
∇
Sufficient condition for local optimality in unconstrained optimization: Any point
x
at which
∇
f(x)
=
0 and
2
f(x)
is positive definite is a local minimum.
∇
∀
x,y
,
∀
λ
∈[
0
,
1
]
,
f (λx
+
(
1
−
λ)y)
≤
λf (x)
+
(
1
−
λ)f (y)
.
A
function
f
is
convex
if
c)
n
if
n
is an even integer,
, 1
/x
,
e
x
. If the Hessian matrix
Common convex functions:
c
,
cx
,
(x
−
|
x
|
exists, it is positive semi-definite.
If
f
is convex and differentiable,
∇
f(
x
)
=
0 if and only if
x
is a global minimum.
