미분적분학실습2-Week-11

912 days ago by jhlee2chn

14.7. Triple Integrals in Cylindrical Coordinates

Ex. 1

r, theta, z = 2, 2*pi/3, 1 r*cos(theta), r*sin(theta), z 
       
(-1, sqrt(3), 1)
(-1, sqrt(3), 1)
T = Cylindrical('height', ['radius', 'azimuth']) T.transform(radius = 2, azimuth = 2*pi/3, height = 1) 
       
(-1, sqrt(3), 1)
(-1, sqrt(3), 1)
x, y, z = 3, -3, -7 r = sqrt(x^2 + y^2) theta = arctan(y/x) r, theta, z 
       
(3*sqrt(2), -1/4*pi, -7)
(3*sqrt(2), -1/4*pi, -7)

Ex. 2

var('r, theta, z') T = Cylindrical('height', ['radius', 'azimuth']) plot3d(r, (r, -2, 2), (theta, 0, 2*pi), transformation = T, opacity = 0.4) 
       
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var('r, theta, z') T = Cylindrical('radius', ['height', 'azimuth']) plot3d(z, (z, -2, 2), (theta, 0, 2*pi), transformation = T, opacity = 0.4) 
       
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Ex. 3

var('x, y, z') u1(x, y) = 1 - x^2 - y^2 p1 = implicit_plot3d(x^2 + y^2 == 1, (x, -1, 1), (y, -1, 1), (z, 0, 4), opacity = 0.4) p2 = plot3d(u1(x, y), (x, -1, 1), (y, -1, 1), opacity = 0.4, color = 'red') p3 = plot3d(0, (x, -1, 1), (y, -1, 1), opacity = 0.4, color = 'orange') p1 + p2 + p3 
       
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var('K, r, theta') f(x, y, z) = K*sqrt(x^2 + y^2) u = r*cos(theta) v = r*sin(theta) w = z integral(integral(integral(f(u, v, w)*r, z, u1(u, v), 4), r, 0, 1), theta, 0, 2*pi) 
       
12/5*pi*K
12/5*pi*K

Ex. 4

var('x, y, z') u1(x, y) = sqrt(x^2 + y^2) p1 = plot3d(u1(x, y), (x, -2, 2), (y, -2, 2), opacity = 0.4) p2 = plot3d(2, (x, -2, 2), (y, -2, 2), opacity = 0.4, color = 'red') p3 = plot3d(0, (x, -2, 2), (y, -2, 2), opacity = 0.4, color = 'orange') p1 + p2 + p3 
       
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var('r, theta') f(x, y, z) =x^2 + y^2 u = r*cos(theta) v = r*sin(theta) w = z integral(integral(integral(f(u, v, w)*r, z, u1(u, v).simplify_full(), 2), r, 0, 2), theta, 0, 2*pi) 
       
16/5*pi
16/5*pi

Exercise 1

var('x, y, z') p1 = implicit_plot3d(x^2 + y^2 == 1, (x, -4, 4), (y, -4, 4), (z, -1, 8), opacity = 0.4) p2 = implicit_plot3d(x^2 + y^2 == 16, (x, -4, 4), (y, -4, 4), (z, -1, 8), opacity = 0.4, color = 'red') p3 = plot3d(y + 4, (x, -4, 4), (y, -4, 4), opacity = 0.4, color = 'green') p4 = plot3d(0, (x, -4, 4), (y, -4, 4), opacity = 0.4, color = 'orange') p1 + p2 + p3 + p4 
       
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var('r, theta') f(x, y, z) = x - y u = r*cos(theta) v = r*sin(theta) w = z integral(integral(integral(f(u, v, w)*r, z, 0, y + 4), r, 1, 4), theta, 0, 2*pi) 
       
0
0

Exercise 2

var('x, y, z') p1 = implicit_plot3d(x^2 + y^2 == 1, (x, -2, 2), (y, -2, 2), (z, 0, 3), opacity = 0.4) p2 = implicit_plot3d(z^2 == 4*x^2 + 4*y^2, (x, -2, 2), (y, -2, 2), (z, 0, 3), opacity = 0.4, color = 'red') p3 = plot3d(0, (x, -2, 2), (y, -2, 2), opacity = 0.4, color = 'orange') p1 + p2 + p3 
       
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var('r, theta') f(x, y, z) = x^2 u = r*cos(theta) v = r*sin(theta) w = z integral(integral(integral(f(u, v, w)*r, z, 0, 2*r), r, 0, 1), theta, 0, 2*pi) 
       
2/5*pi
2/5*pi

14.8. Triple Integrals in Spherical Coordinates

Ex. 1

rho, theta, phi = 2, pi/4, pi/3 rho*sin(phi)*cos(theta), rho*sin(phi)*sin(theta), rho*cos(phi) 
       
(1/2*sqrt(3)*sqrt(2), 1/2*sqrt(3)*sqrt(2), 1)
(1/2*sqrt(3)*sqrt(2), 1/2*sqrt(3)*sqrt(2), 1)
T = Spherical('radius', ['azimuth', 'inclination']) T.transform(radius = 2, azimuth = pi/4, inclination = pi/3) 
       
(1/2*sqrt(3)*sqrt(2), 1/2*sqrt(3)*sqrt(2), 1)
(1/2*sqrt(3)*sqrt(2), 1/2*sqrt(3)*sqrt(2), 1)

Ex. 2

x, y, z = 0, 2*sqrt(3), -2 rho = sqrt(x^2 + y^2 + z^2) phi = arccos(z/rho) theta = arccos(x/(rho*sin(phi))) rho, theta, phi 
       
(4, 1/2*pi, 2/3*pi)
(4, 1/2*pi, 2/3*pi)

Ex. 3

var('x, y, z, r, t, p') f(x, y, z) =exp((x^2 + y^2 + z^2)^(3/2)) u = r*sin(p)*cos(t) v = r*sin(p)*sin(t) w = r*cos(p) integral(integral(integral(f(u, v, w)*r^2*sin(p), r, 0, 1), t, 0, 2*pi), p, 0, pi) 
       
4/3*pi*(e - 1)
4/3*pi*(e - 1)

Ex. 4

var('x, y, z, r, t, p') f(x, y, z) = 1 u = r*sin(p)*cos(t) v = r*sin(p)*sin(t) w = r*cos(p) integral(integral(integral(f(u, v, w)*r^2*sin(p), r, 0, cos(p)), p, 0, pi/4), t, 0, 2*pi) 
       
1/8*pi
1/8*pi

Exercise 1

var('x, y, z') p1 = plot3d(sqrt(x^2 + y^2), (x, -2, 2), (y, -2, 2), opacity = 0.4) p2 = implicit_plot3d(x^2 + y^2 + z^2 == 1, (x, -2, 2), (y, -2, 2), (z, 0, 4), opacity = 0.4, color = 'red') p3 = implicit_plot3d(x^2 + y^2 + z^2 == 4, (x, -2, 2), (y, -2, 2), (z, 0, 4), opacity = 0.4, color = 'orange') p1 + p2 + p3 
       
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solve([x^2 + y^2 + z^2 == 4, z == sqrt(x^2 + y^2)], x, y, z) 
       
[[x == r1, y == -sqrt(-r1^2 + 2), z == sqrt(2)], [x == r2, y ==
sqrt(-r2^2 + 2), z == sqrt(2)]]
[[x == r1, y == -sqrt(-r1^2 + 2), z == sqrt(2)], [x == r2, y == sqrt(-r2^2 + 2), z == sqrt(2)]]
solve(cos(x) == sqrt(2)/2, x) 
       
[x == 1/4*pi]
[x == 1/4*pi]
var('x, y, z, r, t, p') f(x, y, z) = sqrt(x^2 + y^2 + z^2) u = r*sin(p)*cos(t) v = r*sin(p)*sin(t) w = r*cos(p) integral(integral(integral((f(u, v, w)*r^2*sin(p)).simplify_full(), r, 1, 2), p, 0, pi/4), t, 0, 2*pi) 
       
-15/4*pi*(sqrt(2) - 2)
-15/4*pi*(sqrt(2) - 2)

Exercise 2

var('r, t, p') T = Spherical('inclination', ['radius', 'azimuth']) S = Spherical('radius', ['azimuth', 'inclination']) p1 = plot3d(pi/6, (r, 0, 2), (p, 0, 2*pi), transformation = T, opacity = 0.3, color = 'orange') p2 = plot3d(pi/3, (r, 0, 2), (p, 0, 2*pi), transformation = T, opacity = 0.3, color = 'red') p3 = plot3d(2, (t, 0, 2*pi), (p, 0, pi), transformation = S, opacity = 0.3) p1 + p2 + p3 
       
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var('x, y, z, r, t, p') f(x, y, z) = 1 u = r*sin(p)*cos(t) v = r*sin(p)*sin(t) w = r*cos(p) integral(integral(integral((f(u, v, w)*r^2*sin(p)).simplify_full(), r, 0, 2), t, 0, 2*pi), p, pi/6, pi/3) 
       
8/3*pi*(sqrt(3) - 1)
8/3*pi*(sqrt(3) - 1)

14.9. Change of Variables in Multiple Integrals

Ex. 2

var('u, v') T = (u^2 - v^2, 2*u*v) J = jacobian(T, (u, v)).det() print J 
       
4*u^2 + 4*v^2
(u^2 - v^2, 2*u*v)
4*u^2 + 4*v^2
(u^2 - v^2, 2*u*v)
var('x, y') f(x, y) = y integral(integral(f(T[0], T[1])*J, u, 0, 1), v, 0, 1) 
       
2
2

Ex. 3

var('u, v') T = ((u + v)/2, (u - v)/2) J = jacobian(T, (u, v)).det() print J 
       
-1/2
-1/2
var('x, y') f(x, y) = exp((x + y)/(x - y)) integral(integral(f(T[0], T[1])*abs(J), u, -v, v), v, 1, 2) 
       
3/4*(e^2 - 1)*e^(-1)
3/4*(e^2 - 1)*e^(-1)

Exercise 1

var('x, y, u, v') solve([x == 1/4*(u + v), y == 1/4*(v - 3*u)], u, v) 
       
[[u == x - y, v == 3*x + y]]
[[u == x - y, v == 3*x + y]]
Tinv(x, y) = (x - y, 3*x + y) print Tinv(-1, 3), Tinv(1, -3), Tinv(3, -1), Tinv(1, 5) 
       
(-4, 0) (4, 0) (4, 8) (-4, 8)
(-4, 0) (4, 0) (4, 8) (-4, 8)
T = (1/4*(u + v), 1/4*(v - 3*u)) J = jacobian(T, (u, v)).det() print J 
       
1/4
1/4
var('x, y') f(x, y) = 4*x + 8*y integral(integral(f(T[0], T[1])*abs(J), u, -4, 4), v, 0, 8) 
       
192
192

Exercise 2

var('x, y') p1 = implicit_plot(x*y == 1, (x, 0, 4), (y, 0, 3)) p2 = implicit_plot(x*y == 2, (x, 0, 4), (y, 0, 3), color = 'red') p3 = implicit_plot(x*y^2 == 1, (x, 0, 4), (y, 0, 3), color = 'green') p4 = implicit_plot(x*y^2 == 2, (x, 0, 4), (y, 0, 3), color = 'black') p1 + p2 + p3 + p4 
       
var('u, v') solve([u == x*y, v == x*y^2], x, y) 
       
[[x == u^2/v, y == v/u]]
[[x == u^2/v, y == v/u]]
var('u, v') T = (u^2/v, v/u) J = jacobian(T, (u, v)).det() print J 
       
1/v
1/v
var('x, y') f(x, y) = y^2 integral(integral(f(T[0], T[1])*abs(J), u, 1, 2), v, 1, 2) 
       
3/4
3/4