15.4. Green’s Theorem
Ex. 1
var('x, y')
P(x, y) = x^4
Q(x, y) = x*y
dP = diff(P, y)
dQ = diff(Q, x)
integral(integral(dQ - dP, y, 0, 1 - x), x, 0, 1)
1/6 1/6 |
var('t')
A = vector([0, 0])
B = vector([1, 0])
C = vector([0, 1])
r1 = (1 - t)*A + t*B
u, v = r1
du = diff(u, t)
dv = diff(v, t)
c1 = integral(P(u, v)*du + Q(u, v)*dv, t, 0, 1)
print c1
1/5 1/5 |
r2 = (1 - t)*B + t*C
u, v = r2
du = diff(u, t)
dv = diff(v, t)
c2 = integral(P(u, v)*du + Q(u, v)*dv, t, 0, 1)
print c2
-1/30 -1/30 |
r3 = (1 - t)*C + t*A
u, v = r3
du = diff(u, t)
dv = diff(v, t)
c3 = integral(P(u, v)*du + Q(u, v)*dv, t, 0, 1)
print c3
0 0 |
print c1 + c2 + c3
1/6 1/6 |
Ex. 2
var('x, y, r, t')
P(x, y) = 3*y - exp(sin(x))
Q(x, y) = 7*x + sqrt(y^4 + 1)
dP = diff(P, y)
dQ = diff(Q, x)
u = r*cos(t)
v = r*sin(t)
integral(integral((dQ(u, v) - dP(u, v))*r, r, 0, 3), t, 0, 2*pi)
36*pi 36*pi |
Ex. 3
var('x, y, a, b, t')
u = a*cos(t)
v = b*sin(t)
du = diff(u, t)
dv = diff(v, t)
1/2*integral(u*dv - v*du, t, 0, 2*pi)
pi*a*b pi*a*b |
Ex. 4
var('x, y, r, t')
P(x, y) = y^2
Q(x, y) = 3*x*y
dP = diff(P, y)
dQ = diff(Q, x)
f = dQ - dP
u = r*cos(t)
v = r*sin(t)
integral(integral(f(u, v)*r, r, 1, 2), t, 0, pi)
14/3 14/3 |
Ex. 5
var('x, y')
F = vector([-y/(x^2 + y^2), x/(x^2 + y^2)])
P(x, y) = F[0]
Q(x, y) = F[1]
dP = diff(P, y)
dQ = diff(Q, x)
f = (dQ - dP).simplify_full()
print f
0 0 |
var('a, t')
r = vector([a*cos(t), a*sin(t)])
dr = diff(r, t)
integral((F(x = r[0], y = r[1]).inner_product(dr)).simplify_full(), t, 0, 2*pi)
2*pi 2*pi |
Exercise 1
var('x, y')
implicit_plot(4*x^2 + y^2 == 4, (x, -2, 2), (y, -2, 2))
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P(x, y) = 2*x - x^3*y^5
Q(x, y) = x^3*y^8
dP = diff(P, y)
dQ = diff(Q, x)
integral(integral(dQ - dP, y, -sqrt(4 - 4*x^2), sqrt(4 - 4*x^2)), x, -1, 1)
7*pi 7*pi |
var('t')
r1 = vector([cos(t), 2*sin(t)])
parametric_plot(r1, (t, 0, 2*pi))
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u, v = r1
du = diff(u, t)
dv = diff(v, t)
integral(P(u, v)*du + Q(u, v)*dv, t, 0, 2*pi)
7*pi 7*pi |
Exercise 2
A = vector([0, 0])
B = vector([1, 1])
C = vector([0, 1])
line([A, B, C, A])
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var('x, y')
F = vector([sqrt(x^2 + 1), arctan(x)])
P(x, y) = F[0]
Q(x, y) = F[1]
dP = diff(P, y)
dQ = diff(Q, x)
f = (dQ - dP).simplify_full()
print "dQ/dx-dP/dy =", f
integral(integral(f, y, x, 1), x, 0, 1)
dQ/dx-dP/dy = 1/(x^2 + 1) 1/4*pi - 1/2*log(2) dQ/dx-dP/dy = 1/(x^2 + 1) 1/4*pi - 1/2*log(2) |
var('t')
r1 = (1 - t)*A + t*B
dr1 = diff(r1, t)
c1 = integral((F(x = r1[0], y = r1[1]).inner_product(dr1)).simplify_full(), t, 0, 1)
print c1
1/4*pi + 1/2*sqrt(2) + 1/2*arcsinh(1) - 1/2*log(2) 1/4*pi + 1/2*sqrt(2) + 1/2*arcsinh(1) - 1/2*log(2) |
r2 = (1 - t)*B + t*C
dr2 = diff(r2, t)
c2 = integral((F(x = r2[0], y = r2[1]).inner_product(dr2)).simplify_full(), t, 0, 1)
print c2
-1/2*sqrt(2) - 1/2*arcsinh(1) -1/2*sqrt(2) - 1/2*arcsinh(1) |
r3 = (1 - t)*C + t*A
dr3 = diff(r3, t)
c3 = integral((F(x = r3[0], y = r3[1]).inner_product(dr3)).simplify_full(), t, 0, 1)
print c3
0 0 |
expand(c1 + c2 + c3)
1/4*pi - 1/2*log(2) 1/4*pi - 1/2*log(2) |
15.5. Curl and Divergence
Ex. 1
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
F = vector([x*z, x*y*z, -y^2])
print curl(F)
(-x*y - 2*y, x, y*z) (-x*y - 2*y, x, y*z) |
Ex. 3
F = vector([y^2*z^3, 2*x*y*z^3, 3*x*y^2*z^2])
print curl(F)
(0, 0, 0) (0, 0, 0) |
Ex. 4
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*z, x*y*z, -y^2])
print divergence(F)
x*z + z x*z + z |
Exercise 1
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
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^2*y*z, x*y^2*z, x*y*z^2])
print "curl F =", curl(F)
print "div F =", divergence(F)
curl F = (-x*y^2 + x*z^2, x^2*y - y*z^2, -x^2*z + y^2*z) div F = 6*x*y*z curl F = (-x*y^2 + x*z^2, x^2*y - y*z^2, -x^2*z + y^2*z) div F = 6*x*y*z |
Exercise 2
F = vector([exp(x)*sin(y*z), z*exp(x)*cos(y*z), y*exp(x)*cos(y*z)])
print "curl F =", curl(F)
curl F = (0, 0, 0) curl F = (0, 0, 0) |
[P, Q, R] = F
print "fx =", P
print "fy =", Q
print "fz =", R
fx = e^x*sin(y*z) fy = z*cos(y*z)*e^x fz = y*cos(y*z)*e^x fx = e^x*sin(y*z) fy = z*cos(y*z)*e^x fz = y*cos(y*z)*e^x |
# 적분할 때 적분상수 고려해야 함
f = integral(P, x)
print f
print diff(f, y)
print diff(f, z)
e^x*sin(y*z) z*cos(y*z)*e^x y*cos(y*z)*e^x e^x*sin(y*z) z*cos(y*z)*e^x y*cos(y*z)*e^x |
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