Fact Sheet for Midterm 2
•
If
f
(
x, y
) is of class
C
2
then
∂
2
f
∂x∂y
=
∂
2
f
∂y∂x
•
First order Taylor approximation of
f
:
R
n
→
R
at
x
0
:
l
(
x
) =
f
(
x
0
) +
∑
n
i
=1
h
i
∂f
∂x
i
(
x
0
), with
h
=
x

x
0
.
•
Second order Taylor approximation of
f
:
R
n
→
R
at
x
0
:
q
(
x
) =
f
(
x
0
) +
∑
n
i
=1
h
i
∂f
∂x
i
(
x
0
) +
1
2
∑
n
i,j
=1
h
i
h
j
∂
2
f
∂x
i
∂x
j
(
x
0
),
with
h
=
x

x
0
.
•
Differentiation of vectorvalued functions is done component wise.
•
If
c
(
t
) is a path, then its velocity is
v
(
t
) =
c
0
(
t
), its speed is

c
0
(
t
)

,
and its acceleration is
a
(
t
) =
c
00
(
t
).
•
The arc length of a path
c
(
t
) on the interval [
t
0
, t
1
] is
R
t
1
t
0

c
0
(
t
)

dt
.
•
A flow line of a vector field
F
in
R
n
is a path
c
(
t
) in
R
n
such that
c
0
(
t
) =
F
(
c
(
t
)).
•
Let
f
be a function from
R
n
to
R
, then the gradient of
f
is
∇
f
= (
∂f
∂x
1
,
· · ·
,
∂f
∂x
n
).
•
div
F
=
∇·
F
=
∂F
1
∂x
+
∂F
2
∂y
+
∂F
3
∂z
, where
F
= (
F
1
, F
2
, F
3
) is a vector field.
•
curl
F
=
∇ ×
F
=
i
j
k
∂
∂x
∂
∂y
∂
∂z
F
1
F
2
F
3
, where
F
= (
F
1
, F
2
, F
3
) is a vector
field in
R
3
.
• ∇ ×
(
∇
f
) =
0
and
∇ ·
(
∇ ×
F
) = 0 with
f
:
R
3
→
R
, and
F
a vector
field in
R
3
.
•
Let
T
be a linear mapping from
R
2
to
R
2
such that
T
(
x
) =
A
x
, where
A
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 Winter '07
 Enright
 Math, Approximation, Derivative, Vector Calculus, Coordinate system, Spherical coordinate system, Polar coordinate system, order Taylor approximation

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