Ky 0 Kx u, v / 1, u cos v, u sin v n = u 0 0. cfac = Ku cos v 0 K1. r := t/ 1, R cos t, R sin t. K8 cos t sin t 2 C 4 cos t sin t dt = 0

Transcription

1 > restart: with(linalg): with(plots): > #Stokes Theorem Example R:=2: #define vector field F, find curl of F, and parametrize surface (a disc) F:=(x,y,z)->[x*y,y*z,z]; cf:=curl(f(x,y,z),[x,y,z]); r:=(u,v)->[1,u*cos(v),u*sin(v)]; plot1:=plot3d(r(u,v),u=..r,v=..2*pi,axes=boxed): plot2:=fieldplot3d(f(x,y,z),x=-2..2,y=-2..2,z=-2..2,arrows=slim, fieldstrength=fixed,grid=[1,1,1],color=black): plot3:=fieldplot3d(cf,x=-2..2,y=-2..2,z=-2..2,arrows=slim, fieldstrength=fixed,grid=[1,1,1],color=black): #find normal to surface and compute curl of F along surface for flux integral of the curl of F n:=(u,v)->simplify(crossprod(diff(r(u,v),u),diff(r(u,v),v))): n=n(u,v); cfac:=(u,v)->subs(x=r(u,v)[1],y=r(u,v)[2],z=r(u,v)[3],evalm(cf)): cfac=cfac(u,v); #set up the flux integral Int(Int(dotprod(cFAC(u,v),n(u,v),orthogonal),u=..R),v=..2*Pi)=int (int(dotprod(cfac(u,v),n(u,v),orthogonal),u=..r),v=..2*pi); #parametrize boundary curve and set up line integral (or work integral) of vector field along boundary r:=t->[1,r*cos(t),r*sin(t)]; plot4:=spacecurve(r(t),t=..2*pi,color=red,thickness=3,axes=boxed): Int(dotprod(F(r(t)[1],r(t)[2],r(t)[3]),diff(r(t),t),orthogonal),t=..2*Pi)=int(dotprod(F(r(t)[1],r(t)[2],r(t)[3]),diff(r(t),t), orthogonal),t=..2*pi); P:=array(1..2): P[1]:=display({plot1,plot3}): P[2]:=display({plot2,plot4}): display(p); F := x, y, z / x y, y z, z r := cf := Ky Kx u, v / 1, u cos v, u sin v n = u cfac = Ku cos v K1 2 Ku 2 cos v du dv = r := t/ 1, R cos t, R sin t K8 cos t sin t 2 C 4 cos t sin t dt =

2

3 > with(plots): with(linalg): Three given surfaces (and their normals) with the same boundary curve x^2+y^2=r^2 > R:=4: #define 3 surfaces surf1:=(x,y)->[x,y,x+y]; n_surf1:=crossprod(diff(surf1(x,y),x),diff(surf1(x,y),y)); surf2:=(u,v)->[r*cos(u)*cos(v),r*sin(u)*cos(v),r*sin(v)]; n_surf2:=simplify(crossprod(diff(surf2(u,v),u),diff(surf2(u,v),v) )); surf3:=(x,y)->[x,y,1/2*(x^2+y^2)]; n_surf3:=crossprod(diff(surf3(x,y),x),diff(surf3(x,y),y)); # plot surfaces plot1:=plot3d(surf1(x,y),x=-r..r,y=-sqrt(r^2-x^2)..sqrt(r^2-x^2), color=red,scaling=constrained): plot2:=plot3d(surf2(u,v),u=..2*pi,v=..pi,color=green): plot3:=plot3d(surf3(x,y),x=-r..r,y=-sqrt(r^2-x^2)..sqrt(r^2-x^2), color=blue,axes=frame): #define boundary curves b1:=t->[r*cos(t),r*sin(t),r*cos(t)+r*sin(t)]; b2:=t->[r*cos(t),r*sin(t),]; b3:=t->[r*cos(t),r*sin(t),8]; #plot boundary curves b1plot:=spacecurve(b1(t),t=..2*pi,color=red,thickness=5, linestyle=2): b2plot:=spacecurve(b2(t),t=..2*pi,color=green,thickness=5, linestyle=2): b3plot:=spacecurve(b3(t),t=..2*pi,color=blue,thickness=5, linestyle=2): B:=array(1..2): B[2]:=display({b1plot,b2plot,b3plot}): B[1]:=display({plot2,plot1,plot3}): display(b); surf1 := x, y / x, y, x C y surf2 := n_surf1 := K1 K1 1 u, v / R cos u cos v, R sin u cos v, R sin v n_surf2 := 16 cos u cos v 2 16 sin u cos v 2 16 cos v sin v surf3 := x, y / x, y, n_surf3 := Kx Ky x2 C 1 2 y2 b1 := t/ R cos t, R sin t, R cos t C R sin t b2 := t/ R cos t, R sin t, b3 := t/ R cos t, R sin t, 8

4 A given vector field F and curl(f) > F:=(x,y,z)->[-y,x,2]; curlf:=curl(f(x,y,z),[x,y,z]); N:=4: Fplot:=fieldplot3d(F(x,y,z),x=-R..R,y=-R..R,z=-R..2*R,arrows= SLIM,color=gray,grid=[N,N,N],axes=boxed): curlfplot:=fieldplot3d(curlf,x=-r..r,y=-r..r,z=-r..2*r,arrows= SLIM,color=black,grid=[N,N,N],axes=boxed): display({fplot,curlfplot}); F := x, y, z / Ky, x, 2 curlf := 2

5 Stokes Thm: S 7# F, ds = bdd S F, dr Plots > A:=array(1..2): A[1]:=display({plot2,plot1,plot3,curlFplot}): A[2]:=display({b1plot,b2plot,b3plot,Fplot}): plots[display](a);

6 LHS > curlfalongsurface1:=evalm(subs(surf1(x,y)[1]=x,surf1(x,y)[2]= y,surf1(x,y)[3]=z,curlf)); ans1:=int(int(dotprod(curlfalongsurface1,n_surf1,orthogonal), y=-sqrt(r^2-x^2)..sqrt(r^2-x^2)),x=-r..r); evalf(ans1); curlfalongsurface2:=evalm(subs(surf2(u,v)[1]=x,surf2(u,v)[2]= y,surf2(u,v)[3]=z,curlf)); ans2:=int(int(dotprod(curlfalongsurface2,n_surf2,orthogonal), u=..2*pi),v=..pi/2); evalf(ans2); curlfalongsurface3:=evalm(subs(surf3(x,y)[1]=x,surf4(x,y)[2]= y,surf3(x,y)[3]=z,curlf)); ans3:=int(int(dotprod(curlfalongsurface3,n_surf3,orthogonal), y=-sqrt(r^2-x^2)..sqrt(r^2-x^2)),x=-r..r); evalf(ans3); curlfalongsurface1 := 2

7 ans1 := 4 K4 16 K x 2 2 dy dx K 16 K x curlfalongsurface2 := 2 ans2 := 1 32 cos v sin v du dv curlfalongsurface3 := 2 ans3 := 4 K4 K 16 K x 2 16 K x 2 2 dy dx (3.2.1) RHS > v1:=diff(b1(t),t); F1alongb1:=F(b1(t)[1],b1(t)[2],b1(t)[3]); Int(simplify(dotprod(F1alongb1,v1,orthogonal)),t=..2*Pi)=int (simplify(dotprod(f1alongb1,v1,orthogonal)),t=..2*pi); v2:=diff(b2(t),t); F2alongb2:=F(b2(t)[1],b2(t)[2],b2(t)[3]); Int(simplify(dotprod(F2alongb2,v2,orthogonal)),t=..2*Pi)=int (simplify(dotprod(f2alongb2,v2,orthogonal)),t=..2*pi); v3:=diff(b3(t),t); F3alongb3:=F(b3(t)[1],b3(t)[2],b3(t)[3]); Int(simplify(dotprod(F3alongb3,v3,orthogonal)),t=..2*Pi)=int (simplify(dotprod(f3alongb3,v3,orthogonal)),t=..2*pi); 32*Pi=evalf(32*Pi); v1 := K4 sin t, 4 cos t, K4 sin t C 4 cos t F1alongb1 := K4 sin t, 4 cos t, 2 16 K 8 sin t C 8 cos t dt = 3 v2 := K4 sin t, 4 cos t, F2alongb2 := K4 sin t, 4 cos t, 2 16 dt = 3 v3 := K4 sin t, 4 cos t, F3alongb3 := K4 sin t, 4 cos t, 2

8 16 dt = 3 3 = HM--- (1) Redo the above problems where all of the surfaces (surf1, surf2, surf3) are now defined above the circle x^2+y^2=1 (2) Book Problems---see website (3.3.1)