Chapter 1 Chapter 1 Measurment Introduction to Physics Introduction to Vectors Introduction to Calculus( 微微微 Chapter 0 Preface
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Chapter 1Chapter 1 Measurment
Introduction to Physics
Introduction to Vectors
Introduction to Calculus( 微积分 )
Chapter 0 Preface
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
Introduction to Physics
1) Objects studied in physics
2) Methodology for studying physics
3) Some other key points
(See 动画库 \ 力学夹 \ 绪论 .exe)
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
Introduction to Vectors
A scalar is a simple physical quantity that does not depend on direction.
mass, temperature, volume, work…
A vector is a concept characterized by a magnitude and a direction.
force, displacement, velocity…
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
1) Representation of vectors
2) Addition and subtraction of vectors
3) Dot and cross products
(See 动画库 \ 力学夹 \0-4 矢量运算 .exe)
(See 动画库 \ 力学夹 \0-4 矢量运算 .exe)
Chapter 1Chapter 1 Measurment
θ A
B
θ A
B
?
?
Chapter 0 Preface
)(||| θcos|BABA
3.1) Dot product:
θ A
B
θ A
B
)(Bcos
)(
Acos
No problem ， if θ
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
kAjAiAA zyx
kBjBiBB zyx
?BA
)BBB()( kjikAjAiABA zyxzyx
zzyyxx BABABA
zzyyxx BABABABA
ABBA
22 A|AAA
|
CABACBA
)(
Prove it?
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
3.2) Cross product: nABsinBA)(
is a unit vector perpendicular to both and .
, , and also becomes a right handed system. nn
The length of × can be interpreted as the area of the parallelogram having A and B as sides.
A
B
A
B
A
B
A
B
BA
n
BA-AB
θ
AB|BA| ,BA If
0BA ,B//A If
Scalar triple product:
?)( CBA
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
kAjAiAA zyx
kBjBiBB zyx
?BA
)BBB()( kjikAjAiABA zyxzyx
jBABAiBABA zxxzyzzy
)()(
kBABA xyyx
)(
zyx
zyx
BBB
AAA
kji
BA
jBABA zxxz
)( kBABA xyyx
)(
iBABA yzzy
)(
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
Introduction to Calculus( 微积分 )1) Limit of a function
Lxfcx
)(limƒ(x) can be made to be as close to L as desired by making x sufficiently close to c.
“The limit of ƒ of x, as x approaches c, is L."
Note that this statement can be true even if or ƒ(x) is not defined at c. Lcf )(
1
1)(
2
x
xxfExample:
2|1)(lim 11
xx
xxf
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
2) Derivative of a function( 函数的导数 )
• Motion with constant velocity
t
s
t1 t2
12
12 )()()(
tt
tststv
t
s
t1 t2
• Motion with changing speed
12
12 )()()(
tt
tststv
?
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
How to find the instantaneous speed at t1?
12
121
)()(lim)(
12 tt
tststv
tt
• Motion with changing speed
t
tsttstv
t
)()(lim)(
0
tt 1ttt 2
dt
ds
Derivative of s
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
For general function, its derivative is defined as:
x
xfxxf
dx
xdfx
)()(lim
)(0
')(' yxf
x
f(x)
x1 x2
A
A’
tangent
The meaning of derivative of a function:
x
y
tan
1
0limtanlim
xxAA' x
yβ
AA'
)(' 1xftan
tan)(' 1 xf
Chapter 1Chapter 1 Measurment
How big is an infinitesimal?...
0x is infinitesimal.x
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
Example:2xy
xx
xxx
x
xxx
x
xfxxfy
xx
x
22
lim)(
lim
)()(lim'
2
0
22
0
0
Some basic formulae:
0)'( c
xx ee )'(
1)'( xx)( numberrealais
xx cos)'(sin
xx sin)'(cos
xx
1)'(ln
Chapter 1Chapter 1 MeasurmentSome basic rules:
Chapter 0 Preface
'')'( vuvu '')'( uvvuuv
)0(''
)'(2
vv
uvvu
v
u
)(),( xvvfy )(')(')(' xvvfxy
dx
dv
dv
dy
dx
dyor
For a vector:
kdt
dAj
dt
dAi
dt
dA
dt
tAd zyx
)(
')'( CuCu ,C is a const.
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
3) Differential of a function ( 函数的微分 )
If f(x) has its derivative at point x, then f ’(x)dx is its differential at that point.
dxxfdy )('dx
Differential of the function
Differential of the variable
So f ’(x) is also called differential quotient ( 微商 ) dx
dy
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
dvduvud )(udvvduuvd )(
)0()(2
vv
udvvdu
v
ud
CduCud )( ,C is a const.
Operation rule is the same as that for derivative:
......
One application of differential
))((')()( 000 xxxfxfxfy 0xxif
))((')()( 000 xxxfxfxf
00 xWhen xffxf )0(')0()(
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
Example:
0,)0cos(|)'(sin)0sin()sin( 0 xxxxxx x
Following approximate formulae often used in physics ( ) :
xx )sin(
Nxx N 1)1(
xx2
111
xx )1ln(
xex 1
xx )tan(
......
0x
xffxf )0(')0()(
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
4) Integrals( 积分 )
• Motion with constant velocity
• Motion with changing speed
0vtS
t
v
t00
S
t
v
t00
S
How to find S?
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
t
v
t00
t
i
…
ttvttvttvS N )(......)()( 21
NtifttvN
ii ,0,)(
1
0
01
0)()(lim
tN
ii
Nt
dttvttvS
…
In general, the integral from a to b of f(x) with respect to x is expressed as:
b
adxxf )( definite integral
dxxf )( indefinite integral
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
How to find an integral of a function?
)()()( abxdb
a
If function f(x) is continuous on the interval [a, b] and if on the interval (a, b), then )()(' xfx
b
a
b
a
b
adx
dx
xddxxdxxf
)()(')(
)()('),()()( xfxabdxxfb
a
)()(',)()( xfxCxdxxf
Chapter 1Chapter 1 MeasurmentChapter 0 Preface
Example: ?12
1 dx
x
xx
1)'(ln xx ln)(
2ln|ln1 2
1
2
1 xdx
x
Basic integral formulae:
Cxdx
Ckxkdx
Cxxdx sincos
Cxxdx cossin
Cedxe xx
Cxdxx
ln1
)1(,1
1
aCa
xdxx
aa
k,C: const.
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