# MATH 251: Calculus 3, SET8

## 16: Vector Calculus

### 16.9: The Divergence Theorem

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

#### 1. [1145/2]

We use the Divergence Theorem to compute the relevant sum of surface integrals by computing a triple integral over the solid.
(See the Hand Solutions for the tedious surface integral computations.) Remember your JacFacs in triple integrals!
syms r x y z theta
F = [y^2*z^3 2*y*z 4*z^2], div_F = divergence(F, [x y z]) % from 16.5
F = div_F = I = int(int(int(div_F * r, z, r^2, 9), r, 0, 3), theta, 0, 2*pi)
I = format bank
I_appx = double(I)
I_appx = 7634.07
format short
%
figure
fsurf(r*cos(theta), r*sin(theta), r^2, [0 3 0 2*pi], 'g', 'MeshDensity', 12); hold on
fsurf(r*cos(theta), r*sin(theta), 9, [0 3 0 2*pi], 'r', 'MeshDensity', 12)
axis equal; axis([-3 3 -3 3 0 9])
xticks(-3:3:3); yticks(-3:3:3); zticks(0:3:9)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1145/2') #### 2. [1145/4]

Same drill as #1.
syms r x y z theta
F = [x^2 -y z], div_F = divergence(F, [x y z]) % from 16.5
F = div_F = I = int(int(int(div_F * r, x, 0, 2), r, 0, 3), theta, 0, 2*pi)
I = I_appx = double(I)
I_appx = 113.0973
%
figure % The x-axis is stretched to better visualize the cylinder and disks.
fsurf(x, 3*cos(theta), 3*sin(theta), [0 2*pi 0 2], 'y', 'EdgeColor', 'none'); hold on
fsurf(2, r*cos(theta), r*sin(theta), [0 3 0 2*pi], 'r', 'EdgeColor', 'none')
fsurf(0, r*cos(theta), r*sin(theta), [0 3 0 2*pi], 'c', 'EdgeColor', 'none')
axis([0 2 -3 3 -3 3])
xticks(0:2); yticks(-3:3:3); zticks(-3:3:3)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1145/4'); alpha 0.3 #### 3. [1145/6]

As before, we apply the Divergence Theorem to compute the flux through the closed surface, a rectangular box.
syms a b c x y z
F = [x^2*y*z x*y^2*z x*y*z^2], div_F = divergence(F, [x y z]) % from 16.5
F = div_F = I = int(int(int(div_F, x, 0, a), y, 0, b), z, 0, c)
I = #### 4. [1145/8]

We apply the Divergence Theorem to compute the flux through a sphere.
syms x y z rho phi theta
F = [x^3+y^3 y^3+z^3 x^3+z^3], div_F = divergence(F, [x y z]) % from 16.5
F = div_F = div_F = 3*rho^2 % Convert to spherical coordinates.
div_F = I = int(int(int(div_F * rho^2*sin(phi), rho, 0, 2), phi, 0, pi), theta, 0, 2*pi)
I = I_appx = double(I)
I_appx = 241.2743

#### 5. [1145/10]

We apply the Divergence Theorem to compute the flux through a tetrahedron.
syms a b c x y z
F = [z y x*z], div_F = divergence(F, [x y z]) % from 16.5
F = div_F = I = int(int(int(div_F, z, 0, c*(1 - x/a - y/b)), y, 0, b*(1 - x/a)), x, 0, a)
I = #### 6. [1145/12]

We apply the Divergence Theorem to compute the flux through a surface bounded by a circular cylinder and two disks, one circular and one elliptical.
syms r u x y z theta
F = [x*y+2*x*z x^2+y^2 x*y-z^2], div_F = divergence(F, [x y z]) % from 16.5
F = div_F = div_F = 3*r*sin(theta) % Convert to cylindrical coordinates.
div_F = I = int(int(int(div_F * r, z, r*sin(theta)-2, 0), r, 0, 2), theta, 0, 2*pi)
I = I_appx = double(I)
I_appx = -37.6991
%
figure % The x-axis is stretched to better visualize the cylinder and disks.
fsurf(2*cos(theta), 2*sin(theta), u*(2*sin(theta)-2), [0 2*pi 0 1], 'y', 'EdgeColor', 'none')
hold on
fsurf(r*cos(theta), r*sin(theta), r*sin(theta)-2, [0 2 0 2*pi], 'r', 'EdgeColor', 'none')
fsurf(r*cos(theta), r*sin(theta), 0, [0 2 0 2*pi], 'g', 'EdgeColor', 'none')
axis([-2 2 -2 2 -4 0])
xticks(-2:2:2); yticks(-2:2:2); zticks(-4:2:0)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1145/12'); view(54,12); alpha 0.3 #### 7. [1146/14]

We apply the Divergence Theorem to compute the flux through a sphere.
syms a x y z rho phi theta
F = (x^2 + y^2 + z^2)*[x y z],
F = div_F = simplify( divergence(F, [x y z]) ) % from 16.5
div_F = div_F = 5*rho^2 % Convert to spherical coordinates.
div_F = I = int(int(int(div_F * rho^2*sin(phi), rho, 0, a), phi, 0, pi), theta, 0, 2*pi)
I = #### 8. [1146/16]

We apply the Divergence Theorem to compute the flux through a cube.
syms a b c x y z
F = [sin(x) * cos(y)^2 sin(y)^3 * cos(z)^4 sin(z)^5 * cos(x)^6]
F = div_F = divergence(F, [x y z]) % from 16.5
div_F = I = int(int(int(div_F, x, 0, pi/2), y, 0, pi/2), z, 0, pi/2)
I = I_appx = double(I)
I_appx = 2.9300

#### 9. [1146/18]

We apply the Divergence Theorem to compute the flux through a solid bounded by a paraboloid and a circular disk.
Then we subtract off the flux through the bottom circular disk. This leaves the flux through the paraboloid.
For disk, let r . Then F(r(x,y)) . The outward (downward) pointing normal is . So the flux is .
syms r x y z theta
F = [z*atan(y^2) z^3*log(x^2+1) z], div_F = divergence(F, [x y z]) % from 16.5
F = div_F = 1
I = int(int(int(div_F * r, z, 1, 2-r^2), r, 0, 1), theta, 0, 2*pi)
I = I_appx = double(I)
I_appx = 1.5708
%
flux_thru_disk = -sym(pi)
flux_thru_disk = flux_thru_paraboloid = I - (-pi)
flux_thru_paraboloid = %
figure
fsurf(r*cos(theta), r*sin(theta), 2-r^2, [0 1 0 2*pi], 'y', 'MeshDensity', 10); hold on
fsurf(r*cos(theta), r*sin(theta), 1, [0 1 0 2*pi], 'm', 'MeshDensity', 10)
axis equal; axis([-1 1 -1 1 0 3])
xticks(-1:1); yticks(-1:1); zticks(0:3)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1146/18'); alpha 0.5 #### 10. [1146/24]

An outward pointing normal vector to the sphere is n . Let F . Then F n , the scalar field stated in the problem. So the surface integral of the scalar field f over the sphere equals the surface integral of the vector field F over the sphere, which we now compute by applying the Divergence Theorem. Since div F , the integral is the volume of a sphere of radius 1, easily computed using multiplication.
syms x y z
F = [2 2 z], div_F = divergence(F, [x y z])
F = div_F = 1
V = sym(4/3 * pi * 1^3) % V = 4/3 * pi * radius^3
V = V_appx = double(V)
V_appx = 4.1888