>zy=diff(z,y) (x^2-2*x)*(-2*y-x)*exp(-X2-y2-x米y 直接绘制三维曲面 >>[x,y]= meshgrid(-3:2:3,-2:2:2) Z(x.^2-2*x).*exp(-x^2-y.2-x.*y); surf(xyz),axis([-33-22-0.71.5]
>> zy=diff(z,y) zy = (x^2-2*x)*(-2*y-x)*exp(-x^2-y^2-x*y) • 直接绘制三维曲面 >> [x,y]=meshgrid(-3:.2:3,-2:.2:2); >> z=(x.^2-2*x).*exp(-x.^2-y.^2-x.*y); >> surf(x,y,z), axis([-3 3 -2 2 -0.7 1.5])
>> contour(xy,z,30), hold on%绘制等值线 ZX=eXp(-X.^2y.^2-x.*y).*(-2*x+2+2*x3+x.2.*y- 4*x.2-2*x*y) >2y=-x*(x-2)*(2*y+x)*exp(-x^2-y:^2-x*y);%偏导 的数值解 quiver(xyzx,zy)%绘制引力线 .5 . ff、 t1
>> contour(x,y,z,30), hold on % 绘制等值线 >> zx=-exp(-x.^2-y.^2-x.*y).*(-2*x+2+2*x.^3+x.^2.*y- 4*x.^2-2*x.*y); >> zy=-x.*(x-2).*(2*y+x).*exp(-x.^2-y.^2-x.*y); % 偏导 的数值解 >> quiver(x,y,zx,zy) % 绘制引力线
例 已知f(xy,3)=sin(x2y)e-2y=2 求 af(x,y,z) (x20y0x) > syms xy z; fsin(x/2*y)*exp(-x2*y-z/2); > df-difi(diff(diff(f,x, 2),y), z); df-simple(df); pretty rdf -4 z exp(x y-z)(cos(x y)-10 cos(x y)yx+4 242 2422 sin(x y)x y+ 4 cos(x y)x y-sin(x y))
• 例 >> syms x y z; f=sin(x^2*y)*exp(-x^2*y-z^2); >> df=diff(diff(diff(f,x,2),y),z); df=simple(df); >> pretty(df) 2 2 2 2 2 -4 z exp(-x y - z ) (cos(x y) - 10 cos(x y) y x + 4 2 4 2 2 4 2 2 sin(x y) x y+ 4 cos(x y) x y - sin(x y))
多元函数的 Jacobi矩阵 y1=f1(x1,x2,…,xn) 函数y2=(x,2,…,)的Jac0矩阵 ym=fmn(x1,x2,…,xn) dy1/ax ay1/ax2 dy1/axn dylox a 0 yo/ox dy2/a dym/ax1 Oym/ax2.. Oym/axn 格式:J= jacobian(Y,X) 其中,X是自变量构成的向量,Y是由各个函数构成的 向量
• 多元函数的Jacobi矩阵: –格式：J=jacobian(Y,X) 其中，X是自变量构成的向量，Y是由各个函数构成的 向量
651 x=rsin g cos y=rsin 0 sin o, z=rcos 8, 试推导其 Jacobi矩阵 > syms r theta phi; >xr*sin(theta) *cos(phi) >>yr*sin(theta)*sin(phi >>Zr* cos(theta) >>J=jacobian(Ix; y; z[r theta phil) I sin(theta)*cos(phi), rcos(theta)*cos(phi), -*sin(theta) *sin(phi) I sin(theta) *sin(phi), r*cos( theta)*sin(phi), rsin(theta) *cos(phi)l cos(theta) *sin(theta) 0]
• 例： 试推导其 Jacobi 矩阵 >> syms r theta phi; >> x=r*sin(theta)*cos(phi); >> y=r*sin(theta)*sin(phi); >> z=r*cos(theta); >> J=jacobian([x; y; z],[r theta phi]) J = [ sin(theta)*cos(phi), r*cos(theta)*cos(phi), -r*sin(theta)*sin(phi)] [ sin(theta)*sin(phi), r*cos(theta)*sin(phi), r*sin(theta)*cos(phi)] [ cos(theta), -r*sin(theta), 0]