MATH 251: Calculus 3, SET8

16: Vector Calculus

16.7: Surface Integrals

These problems are done with the CAS. See Hand Solutions for details.

1. [1133/8]

The surface integral of a scalar field is computed for a parametric surface.
syms r t u v real
x = 2*u*v, y = u^2-v^2, z = u^2+v^2
x = 
y = 
z = 
f = simplify(x^2 + y^2), s = sqrt(1-u^2)
f = 
s = 
R = [x y z]
R = 
Ru = diff(R,u), Rv = diff(R,v)
Ru = 
Rv = 
n = cross(Ru,Rv)
n = 
mag_n = simplify(sqrt(sum(n.^2)))
mag_n = 
I = int(int(f*mag_n, v, -s, s), u, -1, 1)
I = 
I_appx = double(I)
I_appx = 4.4429
%
R = simplify( subs(R, [u v], [r*cos(t) r*sin(t)]) )
R = 
cone = simplify( R(3)^2 == R(1)^2 + R(2)^2 ) % portion of upper part of z^2 = x^2 + y^2
cone = TRUE
figure
fsurf(R(1), R(2), R(3), [0 1 0 2*pi], 'MeshDensity', 16)
axis equal; axis([-1 1 -1 1 0 1])
xticks(-1:1); yticks(-1:1); zticks(0:1)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1133/8: cone')

2. [1133/12]

The surface integral of a scalar field is computed for a rectangular surface.
syms x y real
f = y, r = [x y 2/3 * (x^(3/2) + y^(3/2))]
f = y
r = 
rx = diff(r,x), ry = diff(r,y)
rx = 
ry = 
n = cross(rx,ry)
n = 
mag_n = simplify(sqrt(sum(n.^2)))
mag_n = 
I = int(int(f*mag_n, x, 0, 1), y, 0, 1)
I = 
I_appx = double(I)
I_appx = 0.7332
%
figure
fsurf(r(3), [0 1 0 1], 'MeshDensity', 12)
axis equal; axis([0 1 0 1 0 1.5])
xticks(0:0.5:1); yticks(0:0.5:1); zticks(0:0.5:1.5)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1133/12: surface')

3. [1133/16]

The surface integral of a scalar field is computed for a portion of a sphere.
syms phi theta positive
x = 1*sin(phi)*cos(theta), y = 1*sin(phi)*sin(theta), z = 1*cos(phi) % spherical coordinates
x = 
y = 
z = 
f = y^2, r = [x y z]
f = 
r = 
rphi = diff(r,phi), rtheta = diff(r,theta)
rphi = 
rtheta = 
n = simplify(cross(rphi,rtheta))
n = 
mag_n = simplify(sqrt(sum(n.^2)))
mag_n = 
I = expand(int(int(f*mag_n, phi, 0, pi/4), theta, 0, 2*pi) )
I = 
I_appx = double(I)
I_appx = 0.2432
%
figure
fsurf(r(1), r(2), r(3), [0 pi/4 0 2*pi], 'MeshDensity', 12)
axis equal; axis([-1 1 -1 1 0 1])
xticks(-1:1); yticks(-1:1); zticks(0:1)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1133/16: surface')

4. [1133/20]

The surface integral of a scalar field is computed for a tin can with its lids.
syms r z theta positive
f = r^2 + z^2 % x^2 + y^2 + z^2
f = 
% 1: cylindrical side
f1 = subs(f, r, 3)
f1 = 
x = 3*cos(theta), y = 3*sin(theta) % cylindrical coordinates
x = 
y = 
R1 = [x y z]
R1 = 
R1theta = diff(R1,theta), R1z = diff(R1,z)
R1theta = 
R1z = 
n1 = simplify(cross(R1theta,R1z))
n1 = 
mag_n1 = simplify(sqrt(sum(n1.^2)))
mag_n1 = 3
I1 = int(int(f1*mag_n1, theta, 0, 2*pi), z, 0, 2)
I1 = 
I1_appx = double(I1)
I1_appx = 389.5575
% 2: bottom circular disk
f2 = subs(f, z, 0)
f2 = 
x = r*cos(theta), y = r*sin(theta) % cylindrical coordinates
x = 
y = 
R2 = [x y 0]
R2 = 
R2r = diff(R2,r), R2theta = diff(R2,theta)
R2r = 
R2theta = 
n2 = simplify(cross(R2r,R2theta))
n2 = 
mag_n2 = simplify(sqrt(sum(n2.^2)))
mag_n2 = r
I2 = int(int(f2*mag_n2, r, 0, 3), theta, 0, 2*pi)
I2 = 
I2_appx = double(I2)
I2_appx = 127.2345
% 3: top circular disk
f3 = subs(f, z, 2)
f3 = 
x = r*cos(theta), y = r*sin(theta) % cylindrical coordinates
x = 
y = 
R3 = [x y 2]
R3 = 
R3r = diff(R3,r), R3theta = diff(R3,theta)
R3r = 
R3theta = 
n3 = simplify(cross(R3r,R3theta))
n3 = 
mag_n3 = simplify(sqrt(sum(n3.^2)))
mag_n3 = r
I3 = int(int(f3*mag_n3, r, 0, 3), theta, 0, 2*pi)
I3 = 
I3_appx = double(I3)
I3_appx = 240.3318
%
I = I1 + I2 + I3
I = 
I_appx = double(I)
I_appx = 757.1238
%
figure
fsurf(R1(1), R1(2), R1(3), [0 2*pi 0 2], 'g', 'EdgeColor', 'none'); hold on
fsurf(R2(1), R2(2), R2(3), [0 3 0 2*pi], 'b', 'EdgeColor', 'none')
fsurf(R3(1), R3(2), R3(3), [0 3 0 2*pi], 'r', 'EdgeColor', 'none')
axis equal; axis([-3 3 -3 3 0 2])
xticks(-3:3); yticks(-3:3); zticks(0:2)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1133/20: tin can with lids')
view(-40,15); alpha 0.4

5. [1133/24]

The surface integral of a vector field across part of a cone (a.k.a., the "flux") is computed with downward orientation.
syms r theta positive
x = r*cos(theta), y = r*sin(theta), z = r % cone in polar coordinates
x = 
y = 
z = r
F = [-x -y z^3], R = [x y z]
F = 
R = 
Rr = diff(R,r), Rtheta = diff(R,theta)
Rr = 
Rtheta = 
n = simplify(cross(Rr,Rtheta)) % Upward orientation OK? NO! We want downward.
n = 
n = -n % So NEGATE normal.
n = 
intg = simplify( dot(F,n) )
intg = 
I = int(int(intg, r, 1, 3), theta, 0, 2*pi)
I = 
I_appx = double(I)
I_appx = -358.5604
%
figure
fsurf(R(1), R(2), R(3), [1 3 0 2*pi], 'MeshDensity', 12)
axis equal; axis([-3 3 -3 3 0 3])
xticks(-3:3); yticks(-3:3); zticks(0:3)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1133/24: frustum of cone')

6. [1133/28]

The surface integral of a vector field across a surface is computed with upward orientation.
syms x y z positive
r = [x y x*sin(y)], F = subs([y*z x*z x*y], z, x*sin(y))
r = 
F = 
rx = diff(r,x), ry = diff(r,y)
rx = 
ry = 
n = cross(rx,ry) % Orientation OK? Yes: it points upward.
n = 
intg = simplify(dot(F,n))
intg = 
I = int(int(intg, x, 0, 2), y, 0, pi)
I = 
I_appx = double(I)
I_appx = 4.9348
%
figure
fsurf(r(1), r(2), r(3), [0 2 0 pi], 'MeshDensity', 14)
axis equal; axis([0 2 0 3 0 2])
xticks(0:2); yticks(0:3); zticks(0:2)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1133/28: surface')