15.6. Parametric Surfaces and Their Areas
Ex. 1
var('u, v')
r = vector([2*cos(u), v, 2*sin(u)])
parametric_plot3d(r, (u, 0, 2*pi), (v, -3, 3), opacity = 0.4)
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Ex. 2
var('u, v')
r = vector([(2 + sin(v))*cos(u), (2 + sin(v))*sin(u), u + cos(v)])
parametric_plot3d(r, (u, 0, 4*pi), (v, 0, 2*pi), opacity = 0.4, mesh = True)
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Ex. 4
var('t, p')
r = vector([2*sin(p)*cos(t), 2*sin(p)*sin(t), 2*cos(p)])
parametric_plot3d(r, (t, 0, 2*pi), (p, 0, pi), opacity = 0.4, mesh =True)
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Ex. 5
var('t, z')
r = vector([2*cos(t), 2*sin(t), z])
parametric_plot3d(r, (t, 0, 2*pi), (z, 0, 1), opacity = 0.4, mesh =True)
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Ex. 7
var('x, y')
r = vector([x, y, 2*sqrt(x^2 + y^2)])
parametric_plot3d(r, (x, -2, 2), (y, -2, 2), opacity = 0.4, mesh = True)
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var('rho, t')
r = vector([rho*cos(t), rho*sin(t), 2*rho])
parametric_plot3d(r, (rho, 0, 2), (t, 0, 2*pi), opacity = 0.4, mesh = True)
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Ex. 8
var('x, t')
f(x) = sin(x)
r = vector([x, f(x)*cos(t), f(x)*sin(t)])
parametric_plot3d(r, (x, 0, 2*pi), (t, 0, 2*pi), opacity = 0.4, mesh = True)
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Ex. 9
var('x, y, z, u, v')
T = vector([x, y, z])
r = vector([u^2, v^2, u + 2*v])
ru = diff(r, u)
rv = diff(r, v)
normal = ru.cross_product(rv).subs(u = 1, v = 1)
P = r.subs(u = 1, v = 1)
print "Tangent Plane:", normal.inner_product(T - P) == 0
p1 = parametric_plot3d(r, (u, -2, 2), (v, -2, 2), opacity = 0.4)
p2 = implicit_plot3d(normal.inner_product(T - P) == 0, (x, -1, 4), (y, -1, 4), (z, -1, 5), opacity = 0.4, color = 'red')
p1 + p2
Tangent Plane: -2*x - 4*y + 4*z - 6 == 0 Tangent Plane: -2*x - 4*y + 4*z - 6 == 0 Sleeping...
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Ex. 10
var('t, p')
r = vector([2*sin(p)*cos(t), 2*sin(p)*sin(t), 2*cos(p)])
rp = diff(r, p)
rt = diff(r, t)
integral(integral((rp.cross_product(rt)).norm(), t, 0, 2*pi), p, 0, pi)
16*pi 16*pi |
Ex. 11
var('x, y, r, t')
f(x, y) = x^2 + y^2
fx = diff(f(x, y), x)
fy = diff(f(x, y), y)
u = r*cos(t)
v = r*sin(t)
integral(integral((sqrt(1 + fx^2 + fy^2).subs(x = u, y = v))*r, r, 0, 3), t, 0, 2*pi)
-1/6*pi + 1/6*(37*sqrt(37))*pi -1/6*pi + 1/6*(37*sqrt(37))*pi |
15.7. Surface Integrals
Ex. 1
var('x, y, z, t, p')
f(x, y, z) = x^2
r = vector([sin(p)*cos(t), sin(p)*sin(t), cos(p)])
rp = diff(r, p)
rt = diff(r, t)
s = rp.cross_product(rt).simplify_full() # 중간과정에서 식을 정리하였음
s1 = (s.inner_product(s)).simplify_full() # 중간과정에서 식을 정리하였음
print s1
sin(p)^2 sin(p)^2 |
integral(integral(f(r[0], r[1], r[2])*sin(p), p, 0, pi), t, 0, 2*pi)
4/3*pi 4/3*pi |
Ex. 2
var('x, y')
g(x, y) = x + y^2
plot3d(g, (x, 0, 1), (y, 0, 2), opacity = 0.4)
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gx = diff(g(x, y), x)
gy = diff(g(x, y), y)
integral(integral(y*sqrt(1 + gx^2 + gy^2), y, 0, 2), x, 0, 1)
13/3*sqrt(2) 13/3*sqrt(2) |
Ex. 3
var('x, y, z')
p1 = plot3d(0, (x, -1, 1), (y, -1, 1), color = 'green', opacity = 0.4)
p2 = implicit_plot3d(x^2 + y^2 == 1, (x, -1, 1), (y, -1, 1), (z, 0, 2), color = 'red', opacity = 0.4)
p3 = plot3d(1 + x, (x, -1, 1), (y, -1, 1), opacity = 0.4)
p1 + p2 + p3
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# S1
var('t')
r = vector([cos(t), sin(t), z])
rt = diff(r, t)
rz = diff(r, z)
rtrz = rt.cross_product(rz)
print rtrz.inner_product(rtrz).simplify_full()
1 1 |
integral(integral(z, z, 0, 1 + cos(t)), t, 0, 2*pi)
3/2*pi 3/2*pi |
# S3
var('r')
g(x, y) = 1 + x
gx = diff(g(x, y), x)
gy = diff(g(x, y), y)
u = r*cos(t)
v = r*sin(t)
integral(integral((((1 + x)*sqrt(1 + gx^2 + gy^2)).subs(x = u, y = v))*r, r, 0, 1), t, 0, 2*pi)
sqrt(2)*pi sqrt(2)*pi |
Ex. 4
var('x, y, z, p, t')
F = vector([z, y, x])
r = vector([sin(p)*cos(t), sin(p)*sin(t), cos(p)])
p1 = plot_vector_field3d(F, (x, -1, 1), (y, -1, 1), (z, -1, 1))
p2 = parametric_plot3d(r, (p, 0, pi), (t, 0, 2*pi), opacity = 0.4)
p1 + p2
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rp = diff(r, p)
rt = diff(r, t)
integral(integral(F(x = r[0], y = r[1], z = r[2]).inner_product(rp.cross_product(rt)), p, 0, pi), t, 0, 2*pi)
4/3*pi 4/3*pi |
Ex. 5
var('x, y, z')
F = vector([y, x, z])
g(x, y) = 1 - x^2 - y^2
p1 = plot_vector_field3d(F, (x, -1, 1), (y, -1, 1), (z, -1, 1))
p2 = plot3d(g, (x, -1, 1), (y, -1, 1), opacity = 0.4)
p3 = plot3d(0, (x, -1, 1), (y, -1, 1), opacity = 0.4, color = 'orange')
p1 + p2 + p3
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# S1
var('r, t')
P, Q, R = F
gx = diff(g(x, y), x)
gy = diff(g(x, y), y)
u = r*cos(t)
v = r*sin(t)
integral(integral((r*((-P*gx - Q*gy + R.subs(z = g(x, y))).subs(x = u, y = v))).simplify_full(), r, 0, 1), t, 0, 2*pi)
1/2*pi 1/2*pi |
# S2
n = vector([0, 0, -1])
print F.inner_product(n).subs(z = 0)
0 0 |
15.8. Stokes’ Theorem
Ex. 1
var('x, y, z')
F = vector([-y^2, x, z^2])
p1 = plot_vector_field3d(F, (x, -1, 1), (y, -1, 1), (z, 1, 3))
p2 = implicit_plot3d(y + z == 2, (x, -1, 1), (y, -1, 1), (z, 1, 3), opacity = 0.4)
p3 = implicit_plot3d(x^2 + y^2 == 1, (x, -1, 1), (y, -1, 1), (z, 1, 3), opacity = 0.4, color = 'red')
p1 + p2 + p3
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var('x, y, z')
def curl(F): # curl 함수를 정의
assert(len(F) == 3) # 3차원 벡터인지 확인
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)])
return curl
print curl(F)
(0, 0, 2*y + 1) (0, 0, 2*y + 1) |
var('r, t')
P, Q, R = curl(F)
g(x, y) = 2 - y
gx = diff(g(x, y), x)
gy = diff(g(x, y), y)
u = r*cos(t)
v = r*sin(t)
integral(integral((r*((-P*gx - Q*gy + R.subs(z = g(x, y))).subs(x = u, y = v))).simplify_full(), r, 0, 1), t, 0, 2*pi)
pi pi |
Ex. 2
var('x, y, z')
F = vector([x*z, y*z, x*y])
p1 = plot_vector_field3d(F, (x, -2, 2), (y, -2, 2), (z, -2, 2))
p2 = implicit_plot3d(x^2 + y^2 + z^2 == 4, (x, -2, 2), (y, -2, 2), (z, -2, 2), opacity = 0.4)
p3 = implicit_plot3d(x^2 + y^2 == 1, (x, -2, 2), (y, -2, 2), (z, -2, 2), opacity = 0.4, color = 'red')
p1 + p2 + p3
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solve([x^2 + y^2 + z^2 == 4, x^2 + y^2 == 1], x, y, z)
[[x == r1, y == -sqrt(-r1^2 + 1), z == -sqrt(3)], [x == r2, y == -sqrt(-r2^2 + 1), z == sqrt(3)], [x == r3, y == sqrt(-r3^2 + 1), z == -sqrt(3)], [x == r4, y == sqrt(-r4^2 + 1), z == sqrt(3)]] [[x == r1, y == -sqrt(-r1^2 + 1), z == -sqrt(3)], [x == r2, y == -sqrt(-r2^2 + 1), z == sqrt(3)], [x == r3, y == sqrt(-r3^2 + 1), z == -sqrt(3)], [x == r4, y == sqrt(-r4^2 + 1), z == sqrt(3)]] |
var('t')
r = vector([cos(t), sin(t), sqrt(3)])
dr = diff(r, t)
integral(F(x = r[0], y = r[1], z = r[2]).inner_product(dr), t, 0, 2*pi)
0 0 |
15.9. The Divergence Theorem
Ex. 1
var('x, y, z')
F = vector([z, y, x])
p1 = plot_vector_field3d(F, (x, -1, 1), (y, -1, 1), (z, -1, 1))
p2 = implicit_plot3d(x^2 + y^2 + z^2 == 1, (x, -1, 1), (y, -1, 1), (z, -1, 1), opacity = 0.4, color = 'red')
p1 + p2
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var ('x, y, z')
def divergence(F): # div 함수를 정의
assert(len(F) == 3) # 3차원 벡터인지 확인
return diff(F[0], x) + diff(F[1], y) + diff(F[2], z)
print divergence(F)
1 1 |
Ex. 2
var('x, y, z')
p1 = plot3d(1 - x^2, (x, -2, 2), (y, -2, 2), opacity = 0.4)
p2 = plot3d(0, (x, -2, 2), (y, -2, 2), opacity = 0.4, color = 'red')
p3 = implicit_plot3d(y == 0, (x, -2, 2), (y, -2, 2), (z, -2, 2), opacity = 0.4, color = 'green')
p4 = implicit_plot3d(y + z == 2, (x, -2, 2), (y, -2, 2), (z, -2, 2), opacity = 0.4, color = 'orange')
p1 + p2 + p3 + p4
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var ('x, y, z')
def divergence(F): # div 함수를 정의
assert(len(F) == 3) # 3차원 벡터인지 확인
return diff(F[0], x) + diff(F[1], y) + diff(F[2], z)
F = vector([x*y, y^2 + exp(x*z^2), sin(x*y)])
print divergence(F)
3*y 3*y |
integral(integral(integral(divergence(F), y, 0, 2 - z), z, 0, 1 - x^2), x, -1, 1)
184/35 184/35 |
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