高等数学英文板总结
-
函数
In mathematics, a
function
is a relation
between a set of inputs and a set of permissible
outputs
with the property that each
input is related to exactly one output.
极限
In
mathematics, a
limit
is the
value that a function or sequence
approaches some value.
The
concept
of
a
limit
of
a
sequence
is
further
generalized
to
the
concept
of
a
limit
of
a
topological
net, and is closely related to limit and direct
limit in category theory.
In formulas, a
limit
is
usually
denoted
as
in
limn
→
c(an)
=
L,
and
the
fact
of
approaching
a
limit
is
represented by the right arrow
(
→
) as in an
→
L.
Suppose
f
is
a
real-valued
function
and
c
is
a
real
number.
The
expression
lim
f
(
x
)
L
x
c
means that f(x) can
be made to be as close to L as desired by making x
sufficiently close to c.
无穷小
Infinitesimal
In
common
speech,
an
infinitesimal
object
is
an
object
which
is
smaller
than
any
feasible
measurement, but
not zero in size; or, so small that it cannot be
distinguished from zero by any
available means.
无穷大
连续函数
In
mathematics, a
continuous function
is, roughly speaking, a function for
which small changes
in the input result
in small changes in the output.
介值定理
In
mathematical analysis, the
intermediate
value theorem
states that if a
continuous function f
with an interval
[a, b] as its domain takes values f(a) and f(b) at
each end of the interval, then it
also
takes
any
value
between
f(a)
and
f(b)
at
some
point
within
the
interval.
This
has
two
important
specializations: If a continuous function has
values of opposite sign inside an interval,
then it has a root in that interval
(Bolzano's theorem).[1] And, the image of a
continuous function
over an interval is
itself an interval.
导数
The
derivative
of
a function of a real variable measures the
sensitivity to change of a quantity (a
function
or
dependent
variable)
which
is
determined
by
another
quantity
(the
independent
variable).
Suppose that x
and y are real numbers and that y is a function of
x, that is, for every value of x,
there
is
a
corresponding
value
of
y.
This
relationship
can
be
written
as
y
=
f(x).
If
f(x)
is
the
equation for a straight
line, then there are two real numbers m and b such
that y = m x + b. m is
called the slope
and can be determined from the formula:
m
chang
in
< br>y
y
，
where
chang
in
x
x
the
symbol
Δ
(the
uppercase form of the Greek letter Delta) is an
abbreviation for
follows
that
Δ
y
=
m
Δ
x.
A
general
function
is
not
a
line,
so
it
does
not
have
a
slope.
The
derivative of f at the point x is the
slope of the linear approximation to f at the
point x.
微分
罗尔定理
In
calculus,
Rolle's
theorem
essentially
states
that
any
real-
valued
differentiable
function
that
attains equal values at two distinct
points must have a stationary point somewhere
between them;
that is, a point where
the first derivative (the slope of the tangent
line to the graph of the function)
is
zero.
If
a
real-
valued
function
f
is
continuous
on
a
closed
interval
[a, b],
differentiable
on
the
open
interval
(a, b),
and
f(a) =
f(b),
then
there
exists
a
c
in
the
open
interval
(a, b)
such
that
f
/
(
c
)
0
. This version of Rolle's theorem is
used to prove the mean value theorem, of which
Rolle's theorem is indeed a special
case. It is also the basis for the proof of
Taylor's theorem.
拉格朗日中值定理
La
grange
’
s mean value theorem
f
(
b
)
f
(
a
)
f
'
(
)
< br>b
a
柯西中值定理
Cauchy's mean value theorem
Cauchy's
mean
value
theorem,
also
known
as
the
extended
mean
value
theorem,
is
a
generalization of the mean
value theorem. It states: If functions f and g are
both continuous on the
closed
interval
[a,b],
and
differentiable
on
the
open
interval
(a,
b),
then
there
exists
some
c
∈
(a,b), such
that
(
f
(
b
)
f
(
a
))
g
'
(
c
)
(
g
(
b
)
g
(
a
))
f
'
(
c
)
；
Of course, if g(a)
≠
g(b) and if
g
′
(c)
≠
0, this is
equivalent to:
f
'
(
c
)
f
(<
/p>
b
)
f
(
a
)
。
g
'
(
c
)
g
(
b
)
g
(
a
)
洛必
达法则
L'Hô
pital's rule
In
calculus,
l'Hô
pital's
rule
(pronounced:
[lopi
ˈ
tal])
uses
derivatives
to
help
evaluate
limits
involving
indeterminate forms. Application of the rule often
converts an indeterminate form to a
determinate form, allowing easy
evaluation of the limit.
In its
simplest form, l'Hô
pital's rule states
that for functions f and g which are
differentiable on an
open
interval
I
except
possibly
at
a
point
c
contained
in
I:
If
lim
x
c
f
(
x
)
lim
g
(
x
)
0
x
c
or
,
and
lim
x
c
f
'
(
x
)
exists,
and
g
'
(
x
)
0
for
all
x
in
I
with
x
≠
c,
g
'
(
x
)
then
lim
x
c
f
(
x<
/p>
)
f
'
(
x
)
lim
.
g
(
x
)
g
'
(
x
)
x
c
The differentiation of
the numerator and denominator often simplifies the
quotient and/or converts
it to a
determinate form, allowing the limit to be
evaluated more easily.
泰勒公式
Taylor's theorem
Statement of
the theorem
The
precise
statement
of
the
most
basic
version
of
Taylor's
theorem
is
as
follows:
Taylor's
k
≥
1 be an integer and let
the function f : R
→
R be k times differentiable at the
point
a
∈
R.
Then
there
exists
a
function
hk :
R
→
R
such
that
This is called the Peano form of the
remainder.
不定积分
Antiderivative
In
calculus,
an
antiderivative,
primitive
integral
or
indefinite
integral[1]
of
a
function
f
is
a
differentiable function F whose
derivative is equal to f, i.e., F
′
= f. The
process of solving for
antiderivatives
is called antidifferentiation (or indefinite
integration) and its opposite operation is
called differentiation, which is the
process of finding a derivative.
定积分
Integration
Through the fundamental theorem of
calculus, which they independently developed,
integration is
connected
with
differentiation:
if
f
is
a
continuous
real-valued
function
defined
on
a
closed
interval
[a,
b],
then,
once
an
antiderivative
F
of
f
is
known,
the
definite
integral
of
f
over
that
interval is given by
多元函数
Functions
with multiple inputs and outputs
The
concept of function can be extended to an object
that takes a combination of two (or more)
argument
values
to
a
single
result.
This
intuitive
concept
is
formalized
by
a
function
whose
domain is the Cartesian product of two
or more sets.
重积分
Multiple
integral
The
multiple
integral
is a generalization of the
definite integral to functions of more than one
real
variable, for example, f(x, y) or
f(x, y, z). Integrals of a function of two
variables over a region in
R2 are
called double integrals, and integrals of a
function of three variables over a region of R3
are called triple integrals.
曲线积分
Line integral
In
mathematics,
a
line
integral
is
an
integral
where
the
function
to
be
integrated
is
evaluated
along
a
curve.
The
terms
path
integral,
curve
integral,
and
curvilinear
integral
are
also
used;
contour integral as
well, although that is typically reserved for line
integrals in the complex plane.
对坐标的曲线积分
Line integral of a
scalar field
For some scalar field f :
U
⊆
Rn
→
R, the line
integral along a piecewise smooth curve C
⊂
U is defined
as
where r: [a, b]
→
C is an arbitrary bijective
parametrization of the curve C such
that r(a) and r(b) give the endpoints of C and .
The function
f
is
called
the
integrand,
the
curve
C
is
the
domain
of
integration,
and
the
symbol
ds
may
be
intuitively interpreted as an
elementary arc length. Line integrals of scalar
fields over a curve C do
not
depend
on
the
chosen
parametrization
r
of
C.
Geometrically,
when
the
scalar
field
f
is
defined over a plane (n=2), its graph
is a surface z=f(x,y) in space, and the line
integral gives the
(signed) cross-
sectional area bounded by the curve C and the
graph of f.
对弧长的曲线积分
Line
integral of a vector field
For a vector
field F : U
⊆
Rn
→
Rn, the line
integral along a piecewise smooth curve C
⊂
U,
in
the
direction
of
r,
is
defined
as