Practice Problems for Exam 4

1(a): Line integral for scalar function
1(b): Work integral for vector field
1(c): Apply fundamental theorem of line integrals
2(a): Work integral for 2D vector field
2(b): Apply Green's theorem
3: Surface integral for scalar function
4(a): Flux surface integral for vector field.
4(b): Apply divergence theorem

1(a): Line integral for scalar function

Let C denote the helix given by (2*cos(t), 2*sin(t), t) for t in [0,2*pi]. Find the integral of (x+y+z) over the curve C.

syms x y z t real
f = x + y + z;
r = [2*cos(t),2*sin(t),t];            % parametrization r(t)
rp = diff(r,t)                        % r'(t)
normrp = simplify(norm(rp))           % ||r'||
fr = subs(f,{x,y,z},r)                % substitute parametrization in f
I = int(fr*normrp,t,0,2*pi)

rp =
[ -2*sin(t), 2*cos(t), 1]
normrp =
5^(1/2)
fr =
t + 2*cos(t) + 2*sin(t)
I =
2*5^(1/2)*pi^2

1(b): Work integral for vector field

Find the work integral x*dx + y*dy + z*dz over the curve C.

F = [x,y,z]                           % vector field
Fr = subs(F,{x,y,z},r)                % substitute parametrization in F
integrand = dot(Fr,rp)
I = int(integrand,t,0,2*pi)           % integrate dot product of F(r(t)) and r'(t)

F =
[ x, y, z]
Fr =
[ 2*cos(t), 2*sin(t), t]
integrand =
t
I =
2*pi^2

1(c): Apply fundamental theorem of line integrals

Show that the vector field from (b) is conservative. Find the integral W using the fundamental theorem of line integrals.

curl(F,[x y z])                           % curl of F is (0,0,0), hence F is conservative
f = potential(F,[x y z])                  % find potential
A = subs(r,t,0)                           % starting point of curve C
B = subs(r,t,2*pi)                        % end point of curve C
I = subs(f,{x,y,z},B) - subs(f,{x,y,z},A) % f(B) - f(A)

ans =
 0
 0
 0
f =
x^2/2 + y^2/2 + z^2/2
A =
[ 2, 0, 0]
B =
[ 2, 0, 2*pi]
I =
2*pi^2

2(a): Work integral for 2D vector field

Let C denote the closed curve in the plane given by x^2+y^2=4, traversed counterclockwise. Find the work integral (x-y)*dx+(x+y)*dy over the curve C.

syms x y t real
F = [x-y,x+y];                       % vector field F
r = [2*cos(t),2*sin(t)];             % parametrization r(t) = (X,Y)
rp = diff(r,t)                       % r'(t)
Fr = subs(F,{x,y},r)                 % substitute parametrization in F
integrand = simplify(dot(Fr,rp))
W = int( integrand,t,0,2*pi)         % integrate dot product of F(r(t)) and r'(t)

rp =
[ -2*sin(t), 2*cos(t)]
Fr =
[ 2*cos(t) - 2*sin(t), 2*cos(t) + 2*sin(t)]
integrand =
4
W =
8*pi

2(b): Apply Green's theorem

Find the integral W from (a) using Green's theorem.

g = diff(F(2),x) - diff(F(1),y)           % Find g = F2_x - F1_y
syms r theta real
X = r*cos(theta); Y = r*sin(theta);       % polar coordinates
gr = subs(g,{x,y},{X,Y})                  % substitute polar coordinates in g
W = int( int( g*r, r,0,2 ), theta,0,2*pi) % integrate g*r*dr*dtheta

g =
2
gr =
2
W =
8*pi

3: Surface integral for scalar function

Let Sigma be the part of the sphere x^2+y^2+z^2=4 with x>=0, y>=0, z>=0. Write the surface integral of x+y+z over Sigma as an iterated integral over theta,phi.

syms x y z phi theta real
f = x + y + z
r = [2*sin(phi)*cos(theta),2*sin(phi)*sin(theta),2*cos(phi)] % parametrization r(phi,theta)
                                    % for phi in [0,pi/2] and theta in [0,pi/2]
rphi = diff(r,phi)                  % find partial derivatives
rtheta = diff(r,theta)
N = simplify(cross(rphi,rtheta))    % find cross product
normN = simplify(norm(N))           % norm of cross product (can omit abs since phi is in [0,pi]

fr = subs(f,{x,y,z},r)              % substitute parametrization in f
integrand = fr*normN                % (can omit abs)
I = int( int( integrand, phi,0,pi/2), theta,0,pi/2 )

f =
x + y + z
r =
[ 2*cos(theta)*sin(phi), 2*sin(phi)*sin(theta), 2*cos(phi)]
rphi =
[ 2*cos(phi)*cos(theta), 2*cos(phi)*sin(theta), -2*sin(phi)]
rtheta =
[ -2*sin(phi)*sin(theta), 2*cos(theta)*sin(phi), 0]
N =
[ 4*cos(theta)*sin(phi)^2, 4*sin(phi)^2*sin(theta), 2*sin(2*phi)]
normN =
4*abs(sin(phi))
fr =
2*cos(phi) + 2*cos(theta)*sin(phi) + 2*sin(phi)*sin(theta)
integrand =
4*abs(sin(phi))*(2*cos(phi) + 2*cos(theta)*sin(phi) + 2*sin(phi)*sin(theta))
I =
6*pi

4(a): Flux surface integral for vector field.

Let Sigma denote the whole sphere x^2+y^2+z^2=4 and consider the vector field F=(x,y,z). Find the flux integral of F over Sigma.

F = [x,y,z]                          % vector field
Fr = subs(F,{x,y,z},r)               % substitute parametrization in F
integrand = simplify(dot(Fr,N))      % dot product of F(r(t)) and cross(rphi,rtheta)
I = int( int( integrand, phi,0,pi), theta,0,2*pi )  % now integrate over whole sphere

F =
[ x, y, z]
Fr =
[ 2*cos(theta)*sin(phi), 2*sin(phi)*sin(theta), 2*cos(phi)]
integrand =
8*sin(phi)
I =
32*pi

4(b): Apply divergence theorem

Evaluate the integral I from (a) using the divergence theorem.

g = divergence(F,[x y z])            % find g = div F
syms rho phi theta real
r = [rho*sin(phi)*cos(theta),rho*sin(phi)*sin(theta),rho*cos(phi)]  % spherical coordinates
gr = subs(g,{x,y,z},r)               % substitute spherical coordinates in g
I = int( int( int( gr*rho^2*sin(phi), rho,0,2 ), phi,0,pi ), theta,0,2*pi)

g =
3
r =
[ rho*cos(theta)*sin(phi), rho*sin(phi)*sin(theta), rho*cos(phi)]
gr =
3
I =
32*pi