西南交通大学限修课数学实验题目及答案七

实验课题七多元函数

设u =2

222,,z u y u ∂∂∂∂在M (1 ,2 ,3)的值. syms x y z u

u=sqrt(x^2+y^2+z^2)

f1=diff(diff(u,x),x)

f2=diff(diff(u,y),y)

f3=diff(diff(u,z),z)

x=1;y=2;z=3;

F1=eval(f1)

F2=eval(f2)

F3=eval(f3)

%F1 =

% 441/1777 %F2 =

% 214/1121

%F3 =

% 107/1121

设⎪⎪⎭

⎫ ⎝⎛=23252sin zy x z xy u ,求z y x u ∂∂∂∂3. syms x y z u

u=x*(y^2)*(z^3)*sin((2*x)/(5*z*(y^2)))

f1=diff(diff(diff(u,x),y),z)

%f1 =

%6*y*z^2*sin((2*x)/(5*y^2*z)) - (12*x*z*cos((2*x)/(5*y^2*z)))/(5*y) - (16*x^3*cos((2*x)/(5*y^2*z)))/(125*y^5*z)

二、求多元函数的极值: 求函数z =sin(x )sin(y )sin(x +y )在 0

x=fminsearch(z,[0,pi/2]);

maxf=eval(z);

max=-maxf

%max =

% 7139/3028

求z=cos(x)+sin(y)－sin(2+x+y)在 –pi

syms x

z='cos(x(1))+sin(x(2))-sin(2+x(1)+x(2))'

[x,min]=fminsearch(z,[-pi,pi])

%min =

% -7139/3028

求函数f (x,y )=e 2x (x +y 2+2y )在0

syms x

f='(x(1)+x(2)^2+2*x(2))*exp(2*x(1))';

x=fminsearch(f,[0,1],[-2,0])

minf=eval(f)

%minf =

% -632/465

三、计算重积分：

⎰⎰⎰⎰+=+42222

d )1(d d d )1(x x

D y y x x y x y x syms x y

f1='(x+1)*(y^2)'

s1=int(int(f1,y,x,x^2),x,2,4)

%s1 =

%357836/105

⎰⎰++=D

y x y x I d d )1ln(22：其中D 为圆：122=+y x 所包围的在第一象限的部分.

syms a r f2='r*log(1+r^2)'

s2=int(int(f2,r,0,1),a,0,pi/2)

%f2 =r*sin(a)*log(1+r^2)

%s2 =log(2) - 1/2

求：()⎰⎰⎰++++Ω

z y x z y x z y x d d d ln 2222

22 Ω：1≤x 2+y 2+z 2≤2. syms a b r

f2='sin(a)*log(1+r^2)*r^2' s2=int(int(int(f2,r,1,2),a,0,pi),b,0,2*pi)

%f2 =sin(a)*log(1+r^2)*r^2

%s2 =32/3*log(5)*pi-32/9*pi-8/3*atan(2)*pi-4/3*pi*log(2)+2/3*pi^2 四、计算曲线积分：

求对弧长的曲线积分：⎰+L ds y x 5

22)(，其中L 是圆周x=a cost, y=a sint (0≤t ≤2pi).

syms a t

x=a*cos(t);y=a*sin(t); f=x^2+y^2;

s=int(f*sqrt(diff(x,t)^2+diff(y,t)^2),t,0,2*pi)

%s3 =2*pi*(a^2)^(11/2)

求对坐标的曲线积分：()

⎰-++L y y x x y x d )(d 22，其中L 为由点A (0,0)到点B (1,1)的曲线23x y =

function s=zxjf(p,q,r,x,y,z,a,b)

syms t

s=int(p*diff(x,t)+q*diff(y,t)+r*diff(z,t),t,a,b);

syms t

x=t;y=t^(2/3);z=0;

syms t

s=int(f*sqrt(diff(x,t)^2+diff(y,t)^2+diff(z,t)^2),t,a,b);

end

syms t

x=exp(t)*cos(t); y=exp(t)*sin(t); z=exp(t);

f=1/(x^2+y^2+z^2);

s4=sxjf(f,x,y,z,0,2)

%s4 =

%-(3^(1/2)*(1/exp(2) - 1))/2

五、解微分方程： 求微分方程x e y y y 22=-'+''的通解 y=dsolve('2*D3y+Dy=y+2*exp(x)') %y =

%C4/exp(t*(1/(6*((29^(1/2)*432^(1/2))/432 + 1/4)^(1/3)) -

(1/432*29^(1/2)*432^(1/2) + 1/4)^(1/3))) - 2*exp(x) +

C5*cos((3^(1/2)*t*(1/(6*((29^(1/2)*432^(1/2))/432 + 1/4)^(1/3)) + ((29^(1/2)*432^(1/2))/432 +

1/4)^(1/3)))/2)*exp(t*(1/(12*((29^(1/2)*432^(1/2))/432 + 1/4)^(1/3)) - ((29^(1/2)*432^(1/2))/432 + 1/4)^(1/3)/2)) +

C6*sin((3^(1/2)*t*(1/(6*((29^(1/2)*432^(1/2))/432 + 1/4)^(1/3)) + ((29^(1/2)*432^(1/2))/432 +

1/4)^(1/3)))/2)*exp(t*(1/(12*((29^(1/2)*432^(1/2))/432 + 1/4)^(1/3)) - ((29^(1/2)*432^(1/2))/432 + 1/4)^(1/3)/2))

求微分方程02sin =++''x y y 满足初始条件1)(,1)(='=ππy y 的特解。

y=dsolve('D2y=-y-sin(2*x)','y(pi)=1,Dy(pi)=1')