舒幼生力學(xué)課件chapter第零章數(shù)學(xué)補(bǔ)充知識(shí)

主講教師 1A -2--1-

x ykz2

a11a12a21a22a31a32元素：a i:行標(biāo);j:列 a21 a22

3

4i i Fy

z z

5a11a12ka21ka22a31a32

a12a22a326 ka13 7

ax x x a31x x x

D

a22 a23 a338

D1

a22

a23

,D2方程組 表述 Di

a33i1,2,3 9例1公比0≤q＜1Saaqaq2aq3aqaaqaq2aq3aS1RAB 2R

11 ?? ?? 矢量的大小稱為矢量的模，記為A或A rGMmrAAB A

AA Ax軸單位矢量Ay軸單位矢量k kAAiAjA

jixAx

AAAAAA AAjAkB

AA?A

k~ k~i

A Akiiki

A

(Ai

B

)e 864442122a(2)對(duì)(2)對(duì) AB規(guī)定0φ

φ

ABAB//A//

A A

(為標(biāo)量 ABBAA AA A) AAAA AAAA A B)

k歸一性

kk(AiAjAk)(BiBjB

AxA

jjjiAjiA

kk

AiB 例題3 Wa

b b

z三

BφA

CABsin ABC

AA(αA)Bα(ABAA

1

2

)1

2

) A1B1A1B2A2B1A2B有ijk; jki;ki有ij

AB Bz例4

M例5力

FIΔl PP0 Il0B

ACAC

A(BC)BB

Bx Cx

Ay

Bz Cz

C(A A,BCCA(B

BAC ACA(BC)(AC)B(A2、3、6、8、10、11、14、15、B22、23、

yy

yy y

x yy(xx)y(x)

yAxΔyy(xΔx)yAxΔy2AxΔxA(Δx)ynxΔysinx(cosΔx1)cosxsinyyex(exy0稱為函數(shù)微分,記成dy

dyy(xdx)y(x)——ysin

yAx dyyAx2 dyA(2xdysinx(cosdx1)cosxsin 簡(jiǎn)書(shū)為Ax cossin tan1 1 dx) dx) 2.yyPyyPQOxxy(x)或yyy(xx)y(x) ydyy(xdx)y(x)tan 例6yAx

yy y2 ysin ysinxcosdx1cosxsindxcos ycos ysin1 1y yy1y2yy

yy1y2y2 yy(u) uu(x)yx

ydydyduyu du u例7yAsin(BxC)

yAsinyx

uBx(Acosu)BABcos(BxC)yAcos

yAsin(x2 yAcos(x)Asin2ytan

ysincosy

(sinx)'cosxsinx(coscos2

cos2yx

k1,2(xk)(xxk1xxk1x(xk1xk1x(xk1xykxk(ax)axln(lnx)x

(y)dyddy dxdxy

(n)

dxn sin cos sin

k,,(ex)(n)

k1,yMyMyPQOxxdyy(x y(x)

y隨xy(x) y(x0)

yMyPQNOxxyyMyPQNOxxy(x0)

y(x0)xy xx0y(0)y(0)sinx (2k1

k x[(2k1)

k x0x0y(xy(x0dyy(x0dxy(x0xx0y(x)y(x0xx00y(x)(xx y(x)A(xx)0A(xx)A(xx)2 A(xx)n

確 級(jí)數(shù)的展開(kāi)系A(chǔ)0y(x0),An

1y(n)

(x0),n0000 n,A(xx

1y(n)(x)(x

x)nsinxcosxex(1x)11ln(1

xxx1x1x1xx0=0 sinxx3!5!7!1

0 前2例題8eixcosxisin例題8

x3 x7sinxx3!5!7!x2 x4 x6cosx12!4!6!x2 x3 x4 x5ex

1x eixcosxisin

A(t)

A(t)i

A

Aj j dAA(tdt) j Ax(tdt) j

Ay(tdt)Ay(t)

Az(tdt)Az(t)dAdAi ykj j)

y(t)

z(t)k?dx(t)dy(t)dz(t)

k

?d2~a ~aid2x(t)i

d2y(t)j

d2z(t)kABAxBxAyByAzBzdd(AB)ABABdtdt

d(AxBx)

d(AyBy)

AB Ay B Bzdd(ABBA

f若F(xf(x)f(xF(xF(xf(x

dF(x)f(x)dxf(x)dxF(x2)F(x1 f(x)dxdF(x)F(x2)F(x1

F(x)f(x)f(x)F(x)yfyfO xF(x)hx f(x)f(x)dxh(x2x1F(x2)F(x1

f(x)f(x)dxF(x):

cosxdxsinxsinxdxcosxexdxex1dxlnxC[A1f1(x)A2f2(x)]dxA1f1(x)dxA2f2(x)dxyOyOxxf(x)dxdl

((dx)2yy=fyy=fOxx ldl

1f(x)2 yy(x1,x2,x3, ,xk

T長(zhǎng)方形的體積V長(zhǎng)寬y(x1dx1,x2, ,xk)y(x1,x2,x3, 1T

T R dyy

y

y k k

T

dTVdpp Φ(r)或Φ(x,y,AA)j A(r)或A(xAA)j它可分解成A(r

A A(r) A(x,y,

zA(r)z

Ay(x,y,Az(x,y,b(x,y,z)dlL(x,y,z)dlb LLx,y,z)dl

(x,(x,y,A(x,y, A(r

A(x,y,z)L

LA(x,y,z) dS

(x,y,z)dS(x,y,z)dS (x,y,z)dSSA(x,y,

A(x