2019年高等数学录播视频及课件-11月1日精讲

⾼主讲老师⼀、⼆元函数的⼏何意义：空间中的⼀⼀、⼆元函数的⼏何意义：空间中的⼀

f(x,y)arcsin(3x2y2

xx

2x2y2xy2

⼀、limf(x0x,y0)f(x0, x 按⼀元函数的⽅⾼阶偏导数：z2z z2z

f

(x,(x,⼀、

y,

B. C. D.⼀、

,则

B. C. D.y yxf

y xy

2x2⼀、设uexcosyvexsiny,则uv zx23xy

⼀、设uexcosyvexsiny,则uv uexsin vexsin zx23xy

z2x3y z3x2y

2132831227⼀、

x3

,(x,y)(0,设f(x,y)x2 , (x,y)(0,1

C. D.⼀、

x3

,(x,y)(0,（4）设f(x,y)x2 , (x,y)(0, B.- C.limf(x0x,y0)f(x0,

D. x303

x2 1⼀、

zx3y23xy3xy1，

2z,2z,2z,2 ⼀、

zx3y23xy3xy1，

2z

2

,2

,2 z3x2y23y3 z2x3y9xy2 2z

2

6xy

2x2z 6xy9y1,yx6xy9y⼀、zf(xx,yy)f(x,zAxByo(dzzdxz ⼀、（1）设z=ln(xy)，则dz 1dx1 1dx1

D.⼀、（1）设z=ln(xy)，则dz 1dx1 1dx1 z 1 xy

D.z 1 xy⼀、已知⼆zf(xy)dzy2dx 2zzsin(xy)xexy，求全微分⼀、已知⼆zf(xy)dzy2dx 2z zy2,z z2y zsin(xy)xexy，求全微分zycos(xy)exy

⼀、已知uxsinyex，求函数u ⼀、（4）已知uxsinyex，求函数u u1yexyu1cosyxexy 12e2,

21cos1e22x

du12e2dx

1cos1e2 ⼀、zf[u(x,y),v(x,zzuz u vzzuz u v⼀、zf[u(x,y),v(x,zzuz u vzzuz u v

dzzduzdv z(udxudy)z(vdxvdy) (zuzv)dx(zuzv)dyu v u v⼀、zf[u(x,y),v(x,zzuz u vzzuz u v（1）⼆元函数：F(x,y)dy

dF(x,y)FxdxFydy

F(x,y,z)dzFxdz ⼀、y设zu2lnv，其中uy

求zx⼀、y设zu2lnv，其中uy

求zx

2ulnv

u2xu v 2xln(3x2y) 3xy2 y2(3x2y)z

2ulnv

x)

u22x

v

2x2

ln(3x2y) y2(3x2⼀、设f(xyxyx2y2xy，(f(x,y)( 2y 2x

⼀、设f(xyxyx2y2xy，( f(x,y)( 2y 2x

D.-xy f(xy,xy)(xy)23xyf(u,v)u3⼀、求x ⼀、求x y 1

x x

x2 y2

x21

x2

x2⼀、x2y2x2y2（1）已知

arctany， ⼀、x2y2x2y2已知

arctany， 令F(x,y)

x2y2x2y2x2x2y2 x2

(2x2yy')

y'xyFx(x,y)

xx2

Fy(x,y)

yx2y2

1xdyFx

xyy

x2

(xyy')xy'x2

y'xy⼀、 yxeyx0，求⼀、 yxeyx0，求F(x,y)yxeyxFey1,F1xey dyey1ey 1 1y'eyxeyy'1eyy'

1⼀、设⽅exy2zez0，z,zz(xy)，

x⼀、设⽅exy2zez0，z,zz(xy)，

x令F(x,yz)exy2z则FyexyFxexyF2 zx zy

ez2,

ez⼀、设⽅程xZxy2z3) x

，⼀、（4）设⽅程xZxy2z3

，)

xF(x,y,z)xZxy2z3 zF

y2

zF

xzln ⼆、⼆定义n0Df(x,y)dlimf(i,i0i

(d⼆⼆、⼆定义n0Df(x,y)dlimf(i,i0i

(d⼆kf(x,y)dkf(x, [f(x,y)g(x,y)]df(x,y)dg(x, f(x,y)df(x,y)df(x, 若f(xyg(xy),则f(xy)dg(x 若mf(xyM,则mf(xy)dDD⼆、⼆定义n

f(x,y)d0

f(i,i)

(dsin(x2y2D

x2y242ln(xy)d [ln(xy)]2 D D, 3x5,0y⼆、⼆定义n

f(x,y)d0

f(i,i)

(d⼆Ie(x2y2 Dx2

(0b ⼆、⼆定义n

f(x,y)dlim0

f(i,i)

(ddxdy⼆Ie(x2y2) Dx2y2

(0ba2 区域D的面 在D ∵0x2y2 1e0ex eae(x2y2)dea2DDabe(x2y2)dabea2D⼆、⼆⼆区域D关于y轴对称1）f(x,y)关于x的奇函数f(x,y)dD2）f(x,y)关于x的偶函数f(x,y)d2f(x, （D1为右边一半区域）区域D关于x轴对称1）f(x,y)关于y的奇函数，则f(x,y)dD2）f(x,y)关于y的偶函数，则f(x,y)d2f(x, （D1为上边一半区域⼆、⼆zf(x,zyzy2(x)yy1(x)Dy2(x)Dy1(x) x

y⼆、⼆重积 3.直角坐标系下⼆重积分的计算c zy2(x)yyzy2(x)yy1(x)Dy2(x)Dy1(x)

Dx1(

x2(xyb x b

S(x)

2(x1(x

f(x,y)dy

VaS(x)dx

2(x 1(x

f(x,y)dy]dxV

f(x,y)dxdy

[2(

f(x,y)dx]dyd 1(y)dD⼆、⼆

1 0dx

f(x,y)dy⼆、⼆

1 0dx

f(x,y)dyyy1

⼆、⼆

⼆、⼆（2

yy2y 2x 2x 0dx 2 1y 01y

f(x,y)dyf(x,y)dx

22dx2

f(x,y)dy⼆、⼆考点6、直角坐标系下的⼆求(x2y)dxdy，其中D是由yx2 xD⼆、⼆考点6、直角坐标系下的⼆求(x2y)dxdy，其中D是由yx2 xDxxyyxy

110(x2y)dxdy 0D

(x2 (0,0),

1 x

22 [(y2y6)y(y2y2)]dy0

1(x2y)dxdy 1

x(x2 D

x1[x2 x2)1(xx4)]dx33x ⼆、⼆求Ix2ydxdy，其中D是 yx和yD⼆、⼆（2）求Ix2ydxdy，其中D是 yx和yD⼆、⼆求Ix2ydxdy，其中D是 yx和yDyDox1xyDox1xyyxy

(0,0), I

1dy x2ydxx 2Ix 2

y

1 1 x2y2]x

y dyy x 051

1(x4x6)dx

1

1(y2y4)dy 35 35⼆、⼆求Iyd ，其中D是由yx4和y2D⼆、⼆（3）求Iyd ，其中D是由yx4和y2D

两曲线交点yyx4(8,4)y2ox(2,2)yyx4(8,4)y2ox(2,2)y

2 2I0dx2

ydy

2dxx

ydy280dx[x1(x4)2]dx28 1(644)1(648)18 ⼆、⼆求Iyd ，其中D是由yx4和y2D

两曲线交点yyx4(8,4)y2ox(2,2)yyx4(8,4)y2ox(2,2)y

4I4 4

ydy ydx2y22(y4) 2ydy 18⼆、⼆求x2ey2dxdy，其中D是由x0y1和yD⼆、⼆（4）求x2ey2dxdy，其中D是由x0y1和yD三条直线的交点为y1yyy1yyox 1I 10

yx2ey20 1 3y2]ydy

1y3ey20[3x

316611y2d(ey2)66

y2ey

11ey2d(y26 601 ⼆、⼆3.极坐标系下⼆重积分的计算f(x,y)dxdy

f(rcos,rsin)rdrd

2( 1(

f(rcos,rsin ⼆、⼆3.极坐标系下⼆重积分的计算y

r

idc

rDo

iA⼆、⼆3.极坐标系下⼆重积分的计算r1 r2D f(x,y)dxdy

f(rcos,rsin)rdrd

d2(

f(rcos,rsin ⼆、⼆求ex2y2D

⼆、⼆求ex2y2D

D:0ra,0ex2y2dxdy22 2daer2rdr

d

1er2d(r202

00

(1ea2⼆、⼆求x2y2)d，其中D1x2y24.D⼆、⼆（2）求x2y2)d，其中D1x2y24.DD 02 1ryo12x(x2y2)yo12xDr3drdD 2d0

r3dr1 2

(2

14)d21515 ⼆、⼆4x24x2y2求ID

，其中D为x2y2⼆、⼆4x24x2y2D

，其中D为x2y2sin3dsin2dcosD 0r2cos

(1cos2)dcosID

4r2rdryryr2Do2x 2

4r2r2 2 (4r2)2]2cos 3 32

(1sin3)d83⼆、⼆sin( x2sin( x2y2x2y求ID

dxdy，其中D

1x2y2⼆、⼆sin( x2