[한빛아카데미] 최신 공학수학 with Sage 실습실

예제 1-1

Sage 코딩

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var('t, x, y')
F=vector([x^2+2*x*y, x^2+y^4])
r=vector([t, sin(pi/2*t)])
dr=diff(r,t)
F_dr=F(x=r[0], y=r[1])*dr
integral(F_dr, t, 0, 1)

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예제 1-2

Sage 코딩

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var('t, x, y, z')
F=vector([z-y, x-z, x-y])
r=vector([cos(t), sin(t), 2-cos(t)+sin(t)])
dr=diff(r,t)
F_dr=F(x=r[0], y=r[1], z=r[2])*dr
integral(F_dr, t, 0, 2*pi)

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예제 1-3

Sage 코딩

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var('t, x, y')
F=vector([-y/(x^2+y^2), x/(x^2+y^2)])
r=vector([cos(t), sin(t)])
dr=diff(r,t)
F_dr=F(x=r[0], y=r[1])*dr
integral(F_dr, t, 0, 2*pi)

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예제 2-1

Sage 코딩

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var('x, y')
f=1-x/4-y/3
print integral(integral(f, x, -2, 2), y, -1, 1) # x 먼저, y 나중
print integral(integral(f, y, -1, 1), x, -2, 2) # y 먼저, x 나중

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예제 2-2

Sage 코딩

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var('x, y')
f=3-x-y
print integral(integral(f, y, 0, x), x, 0, 1)

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예제 2-3

Sage 코딩

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var('a, b, x, y, rho')
Ix=integral(integral(rho*y^2, y, 0, -b/a*x+b), x, 0, a)
Iy=integral(integral(rho*x^2, y, 0, -b/a*x+b), x, 0, a)
print "Ix=", Ix
print "Iy=", Iy

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예제 2-4

Sage 코딩

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var('u, v')
f(u, v)=exp(u/v)
1/2*integral(integral(f, u, -v, v), v, 0, 2)

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예제 2-5

Sage 코딩

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var('r, t')
f(r, t)=1/2*r*(r^2*cos(t)^2+1/4*r^2*sin(t)^2)
integral(integral(f, r, 0, 1), t, 0, 2*pi)

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예제 3-1

Sage 코딩

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var('x, y')
M=x^2*y;N=x*y^2
A=diff(N, x)-diff(M, y)
integral(integral(A, y, 0, 1-x), x, 0, 1)

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예제 4-1

Sage 코딩

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var ('x, y, z, rho, phi, theta')
T = Spherical('radius', ['azimuth', 'inclination']) # 구면좌표계
[u, v, w]=T.transform(radius=rho, azimuth=theta, inclination=phi)
F=vector([x^2, y^2, z^2]) # 벡터장
p1=plot_vector_field3d(F, (x, -2, 2), (y, -2, 2), (z, -2, 2))
p2=plot3d(2, (theta, 0, 2*pi), (phi, 0, pi), transformation=T, opacity=0.4)
p1+p2

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Sage 코딩

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[u0, v0, w0]=[u.subs(rho=2), v.subs(rho=2), w.subs(rho=2)] # p = 2일 때
r(theta, phi)=[u0, v0, w0] # 곡면의 매개변수 방정식
n=(diff(r(theta, phi), phi)).cross_product(diff(r(theta, phi), theta)) # 법선벡터
integral(integral(F.subs(x=u0, y=v0, z=w0).inner_product(n), phi, 0, pi), theta, 0, 2*pi) # 이중적분 계산

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예제 5-1

Sage 코딩

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var ('x, y, z')
f(x, y, z)= x*y*z
integral(integral(integral(f, z , 0 , 2- 2*x - y), y, 0 , 2 - 2*x), x, 0, 1)

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예제 5-3

Sage 코딩

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var('x, y, z')
def Div(F):
    assert(len(F) == 3)
    return (diff(F[0], x)+diff(F[1], y)+diff(F[2], z))
​
F=vector([x*y, y*z, z*x])
Div(F)

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Sage 코딩

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var ('x, y, z')
f(x, y, z)=x + y + z
integral(integral(integral(f, x, 0, 2), y, -1, 3), z, 1, 4)

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예제 6-1

Sage 코딩

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var('x, y, z')
r(x, y)=[x, y, 4-x-y/2]
F(x, y)=[x*z, -y, x^2*y]
r_x=diff(r(x, y), x);r_y=diff(r(x, y), y)
n=r_x.cross_product(r_y)
curl=vector([diff(F[2], y) - diff(F[1], z), diff(F[0], z) - diff(F[2], x), diff(F[1], x) - diff(F[0], y)])
integral(integral(curl.inner_product(n), y, 0, 8-2*x), x, 0, 4)

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Sage 코딩

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t=var('t')
para=parametric_plot((t, t^2),(t,0,1));
show(para, aspect_ratio=1) 
integral(t,t,0,1)

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Sage 코딩

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t=var('t')
para=parametric_plot((t^2, t),(t,0,1));
show(para, aspect_ratio=1) 
integral(2*(t^2)*(t)*(2*t)+(t^2)^2,t,0,1)

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Sage 코딩

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t=var('t')
half_circle=parametric_plot((cos(t),sin(t)),(t,0,pi()));
show(half_circle, aspect_ratio=1) 
integral((cos(t)*sin(t)^2+cos(t)^2*sin(t))*sqrt(diff(cos(t),t)^2+diff(sin(t),t)^2),t,0,pi())

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Sage 코딩

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var('t, x, y')
F=vector([3*x+2*x*y, x^2-5*y^2])
r=vector([t, t])
dr=diff(r,t)
F_dr=F(x=r[0], y=r[1])*dr
integral(F_dr, t, 0, 1)

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Sage 코딩

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var('t, x, y')
F=vector([-x*y, x^2])
r=vector([2*cos(t), -5*sin(t)])
dr=diff(r,t)
F_dr=F(x=r[0], y=r[1])*dr
integral(F_dr, t, 0, pi)

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Sage 코딩

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var('t, x, y')
F=vector([-2*x/(x^2+y^2), 3*y/(x^2+y^2)])
r=vector([t, 1-t])
dr=diff(r,t)
F_dr=F(x=r[0], y=r[1])*dr
integral(F_dr, t, 0, 1)

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Sage 코딩

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var('t, x, y')
F=vector([y/(x^2+y^2)^(3/2), -x/(x^2+y^2)^(3/2)])
r=vector([1+3*t, 1+3*t])
dr=diff(r,t)
F_dr=F(x=r[0], y=r[1])*dr
integral(F_dr, t, 0, 1)

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Sage 코딩

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var('x, y')
F=(x-3*y)^2
integral(integral(F, x, 0, y), y, 0, 1)

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Sage 코딩

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var('x, y')
F=3*x-5*y+7
integral(integral(F, y, 2*x^3, x^2), x, 0, 1/2)

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Sage 코딩

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var('x, y')
F=x^2*y
integral(integral(F, y, x^2, 4), x, -2, 2)

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Sage 코딩

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var('x, y')
F=1/4*(8-x-2*y)
integral(integral(F, y, 0, 1), x, 0, 1)

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Sage 코딩

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var('x, y')
F=sqrt(x)+sqrt(y)
integral(integral(F, y, 0, 1), x, 0, 1)

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Sage 코딩

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var('r, t')
F=4-r^2
integral(integral(F*r, r, 0, 2), t, 0, 2*pi)

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Sage 코딩

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var('r, t, a')
F=2*r*cos(t)
2*integral(integral(F*r, r, 0, a), t, -pi/2, pi/2)

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Sage 코딩

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var('x, y')
f(x, y)=sqrt(1-x^3)
integral(integral(f, y, 0, x^2), x, 0, 1)

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Sage 코딩

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var('r, t')
integral(integral(r, r, 0, 2-2*sin(t)), t, 0, 2*pi)

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Sage 코딩

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var('r, t')
integral(integral(r^2*sin(2*t)*r, r, 0, 1), t, 0, pi/2)

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Sage 코딩

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var('r, t')
integral(integral((2-2*r^2)*r, r, 0, cos(t)), t, 0, pi/2)

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Sage 코딩

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var('x, y')
F=exp(-y^2)
integral(integral(F, y, x, 1), x, 0, 1)

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Sage 코딩

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var('r, t')
F=-4*r*sin(t)-9*r^2*sin(t)^2
integral(integral(F*r, r, 1, 2), t, 0, 2*pi)

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Sage 코딩

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var('x, y')
F=7*x
integral(integral(F, x, -2, 2), y, -2, 2)

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Sage 코딩

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var('x, y, z, t')
p_1=implicit_plot3d(y==x^2, (x,0,1), (y, 0,1), (z, 0,4),color="red", opacity=0.4);
p_2=implicit_plot3d(z==2, (x,0,1), (y, 0,1), (z, 0,4),color="yellow", opacity=0.6);
p_3=parametric_plot3d((t, t^2, 2), (t, 0, 1), thickness=5)
show(p_1+p_2+p_3)
integral(x^2*x^2+(x-2)*2*x, x, 0, 1)

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Sage 코딩

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var('t')
-1/2*integral((t-sin(t))*sin(t)-(1-cos(t))^2, t, 0, 2*pi)

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Sage 코딩

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var('t')
a=1;b=1 # a와 b의 값을 변화시키면서 그림을 살펴볼 수 있음
f(t)=(3*a*t)/(1+t^3)
g(t)=(3*b*t^2)/(1+t^3) 
ANSWER=(1/2)*integral(diff(g,t)-diff(f,t),t,0, 2*pi)
print ANSWER
parametric_plot((f(t), g(t)),(t, 0, 2*pi),color="green")

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Sage 코딩

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var('x, y')
F=2*x*y-2*x^2-3*x-2*y+6
3*integral(integral(F, y, 0, 3-x), x, 0, 3)

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Sage 코딩

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var('u, v')
r(u, v)=[v, u*sin(v), u*cos(v)]
n=(diff(r(u, v), u)).cross_product((diff(r(u, v), v)))
nn=norm(n).simplify_trig()
print nn

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var('u, v')
integral(u^3*sqrt(1+u^2), u, 0, 1)*integral(sin(v)^2*cos(v), v, 0, pi)

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Sage 코딩

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var('x, z')
integral(integral(x*z/sqrt(16-x^2)+x, x, 0, 4), z, 0, 4)

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Sage 코딩

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var ('x, y, z, rho, phi, theta, a, b, c, d')
T = Spherical('radius', ['azimuth', 'inclination'])  # 구면 좌표계
[u, v, w]=T.transform(radius=rho, azimuth=theta, inclination=phi)
F=vector([a*x, b*y, c*z])     # 벡터장
[u0, v0, w0]=[u.subs(rho=d), v.subs(rho=d), w.subs(rho=d)] # rho=d일 때
r(theta, phi)=[u0, v0, w0]    # 곡면의 매개변수 방정식
n=(diff(r(theta, phi), phi)).cross_product(diff(r(theta, phi), theta)) # 법선벡터
integral(integral(F.subs(x=u0, y=v0, z=w0).inner_product(n), phi, 0, pi), theta, 0, 2*pi)   # 이중적분 계산

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Sage 코딩

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var ('x, y, z')
integral(integral(integral(2*x*y^2, z, 2, 5), y, 0, 1), x, 2, 3)

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Sage 코딩

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var ('x, y, z')
integral(integral(integral(exp(2*x-y+z), z, 0, x+y), y, 0, x), x, 0, 1)

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Sage 코딩

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var ('r, t, z, a')
integral(integral(integral(r*z, z, 1-cos(t), 1+cos(t)), r, 0, a), t, 0, pi/2)

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Sage 코딩

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var('x, y, z')    
def Div(F):
    assert(len(F) == 3)
    return (diff(F[0], x)+diff(F[1], y)+diff(F[2], z))
​
F=vector([sin(x), 2-y*cos(x), x*y])
Div(F)

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Sage 코딩

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var('x, y, z')    
def Div(F):
    assert(len(F) == 3)
    return (diff(F[0], x)+diff(F[1], y)+diff(F[2], z))
​
F=vector([x^2-3*y*z, -2*x^2*y, 5*z])
Div(F)

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Sage 코딩

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var('x, y, z, t')
p1 = implicit_plot3d(x^2+y^2==4, (x, -2, 2), (y, -2, 2), (z, -5, 5), opacity=0.2, color="red", mesh=True);
p2 = implicit_plot3d(z==0, (x, -2, 2), (y, -2, 2), (z, -5, 5), opacity=0.3, color="blue", mesh=True);
p3 = implicit_plot3d(z==3, (x, -2, 2), (y, -2, 2), (z, -5, 5),   opacity=0.5, color="orange", mesh=True); 
show(p1+p2+p3, aspect_ratio=1)

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var('x, y, z, t, r')
def Div(F):
    assert(len(F)==3)
    return (diff(F[0], x)+diff(F[1], y)+diff(F[2], z))
f=Div([4*x, -2*y^2, z^2])
g(r, t, z)=f.subs(x=2*cos(t), y=2*sin(t), z=z)
integral(integral(integral(g*r, r, 0, 2), t, 0, 2*pi), z, 0, 3)

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Sage 코딩

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var('x, y, z')    
def Div(F):
    assert(len(F) == 3)
    return (diff(F[0], x)+diff(F[1], y)+diff(F[2], z))
​
F=vector([-x^2*y, y*z^2, x*y*z])
Div(F)

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Sage 코딩

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var('x, y, z')    
def Div(F):
    assert(len(F) == 3)
    return (diff(F[0], x)+diff(F[1], y)+diff(F[2], z))
​
A=vector([x^2, y^2, z^2])
Div(A)

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Sage 코딩

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var('x, y, z')    
def Div(F):
    assert(len(F) == 3)
    return (diff(F[0], x)+diff(F[1], y)+diff(F[2], z))
​
F=vector([3*y, y*z, -x*y*z^5])
Div(F)

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Sage 코딩

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var('x, y, z')    
def Div(F):
    assert(len(F) == 3)
    return (diff(F[0], x)+diff(F[1], y)+diff(F[2], z))
​
A=vector([x^3-y*z, -2*x^3*y, 2*y])
Div(A)

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Sage 코딩

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var('x, y, z')    
r(x, y)=[x, y, 1-x-y]
F(x, y, z)=[y^2+z^2, x^2+z^2, x^2+y^2]
r_x=diff(r(x, y), x);r_y=diff(r(x, y), y)
n=r_x.cross_product(r_y)
curl=vector([diff(F[2], y) - diff(F[1], z), diff(F[0], z) - diff(F[2], x), diff(F[1], x) - diff(F[0], y)]) 
print curl.inner_product(n)

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Sage 코딩

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var('x, y, z, u, v')    
r(u, v)=[u*cos(v), u*cos(v), u*sin(v)]
r_u=diff(r(u, v), u);r_v=diff(r(u, v), v)
n=r_u.cross_product(r_v)
F(x, y, z)=[x^2*y^3, 1, z]
curl=vector([diff(F[2], y) - diff(F[1], z), diff(F[0], z) - diff(F[2], x), diff(F[1], x) - diff(F[0], y)]) 
curl_uv=curl.subs(x=u*cos(v), y=u*cos(v), z=u*sin(v))
print n
print curl_uv
curl_uv.inner_product(n)

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Sage 코딩

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var('x, y, z, t')
p1 = implicit_plot3d(x^2+y^2+z^2==4, (x, -2, 2), (y, -2, 2), (z, 0, 2), opacity=0.2, color="red", mesh=True);
p2 = implicit_plot3d(x^2+y^2==1, (x, -2, 2), (y, -2, 2), (z, 0, sqrt(3)), opacity=0.5, color="blue", mesh=True);
p3 = plot3d(0, (x, -2, 2), (y, -2, 2), opacity=0.3, color="orange", mesh=True); 
show(p1+p2+p3, aspect_ratio=1)

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var('i, j, k')
r_t = vector(SR, [cos(t), sin(t), sqrt(3)]);
dr_t = diff(r_t, t);
F = vector(SR, [x, y, x*y]);
F_r_t = vector(SR, [cos(t), sin(t), cos(t)*sin(t)]);
print (F_r_t)                              # F(r(t))
print integral(F_r_t*dr_t, (t, 0, 2*pi))      # 적분 계산

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Sage 코딩

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var('x, y, z, u, v')    
r(u, v)=[u*cos(v), u*sin(v), u*sin(v)+1]
r_u=diff(r(u, v), u);r_v=diff(r(u, v), v)
n=r_u.cross_product(r_v)
F(x, y, z)=[4*x*z, -2*x*y, 2*x*z]
curl=vector([diff(F[2], y) - diff(F[1], z), diff(F[0], z) - diff(F[2], x), diff(F[1], x) - diff(F[0], y)]) 
curl_uv=curl.subs(x=u*cos(v), y=u*sin(v), z=u*sin(v)+1)
print n
print curl_uv
curl_uv.inner_product(n)

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Sage 코딩

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var('x, y, z')    
r(x, y)=[x, y, 4*(1-1/2*x-1/3*y)]
F(x, y, z)=[-4*x*y+2*x*z, 5*x*y-3*y*z, z^2-2*x*z]
r_x=diff(r(x, y), x);r_y=diff(r(x, y), y)
n=r_x.cross_product(r_y)
curl=vector([diff(F[2], y) - diff(F[1], z), diff(F[0], z) - diff(F[2], x), diff(F[1], x) - diff(F[0], y)]) 
curl_xy=curl.subs(z=4*(1-1/2*x-1/3*y))
print n
print curl_xy
curl_xy.inner_product(n)

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Sage 코딩

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var('x, y, z')    
r(x, y)=[x, y, 3-3*x+3*y]
F(x, y, z)=[2*x+y^2, -2*y+z^2, 3*z-x^2]
r_x=diff(r(x, y), x);r_y=diff(r(x, y), y)
n=r_x.cross_product(r_y)
curl=vector([diff(F[2], y) - diff(F[1], z), diff(F[0], z) - diff(F[2], x), diff(F[1], x) - diff(F[0], y)]) 
curl_xy=curl.subs(z=3-3*x+3*y)
print n
print curl_xy
curl_xy.inner_product(n)

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Chapter 10 벡터적분

예제 1-1

예제 1-2

예제 1-3

예제 2-1

예제 2-2

예제 2-3

예제 2-4

예제 2-5

예제 3-1

예제 4-1

예제 5-1

예제 5-3

예제 6-1

연습문제