Embed Size (px)
Citation preview
Math229 Calculus III for Engineers and Scientists
Application to Engineering and Physics
T. SakaiViterbi School of Engineering
University of Southern California
2020
Chapter 10 Vectors
1. (Statics) If all forces Fi (i = 1, · · · , n) acing on a point in a physical system are in balance
n∑i
Fi = 0,
then the system is at an equilibrium (static) state.
Consider a 100-lb weight hangs from two wires as shown below. In this case, all forces (wire tensions T1
and T2 and the weight W = −100 j) are statically in balance:
T1 + T2 + W = 0.
(a) Write T1 and T2 in terms of standard basis vectors (e.g., T1 = |T1| cos(π/6) i + |T1| sin(π/6) j andT2 =? i+? j).
(b) By plugging the expressions of T1 and T2 into the force balance equation, find the tension forcesT1 and T2 and their magnitudes.
30° 30° T T
0.3 m θ = 75°
80 N
r
F
15°
v0
x
y
O α
d
λ
r
100
30° 60°
T1 T2
x
y
2. (Mechanical Engineering) The geometrical arrangement of aircraft control cable and support pulleys isshown below. The use of pulleys allows the cable tension to maintain uniform along the cable. If thecable tension is 50 N, find the magnitudes of the resultant forces acting on the pulleys A and B. (Hint:Force on pulley A is FA = T1 + T2, where T1 = −50 i, T2 =? i+? j. Also note the cable tension isuniform, that is, |T1| = |T2| = |T3| = |T4| = 50 N.)
9000-lb
75°
45°
45°
T1
T2
T3
T4
A
B x
y
cable
3. (Mechanical work) A constant force given by F = 10 i + 18 j− 6 k moves an object along a straight linefrom the point (2, 3, 0) to the point (4, 9, 5). Find the work done, if the distance is measured in metersand the magnitude of the force is measured in newtons.
4. (Aeronautics) The helicopter is in climbing flight along a straight path with 9000-lb of thrust force havingan angle of 75 degrees relative to the flight path as shown. Suppose the helicopter advanced 3000-ft alongthe path. By applying the knowledge of the dot product, find the work done by the helicopter.
5. (Torque) In order to measure the tendency of a body to rotate about the origin, we define a quantitycalled torque. For example, if we tighten a bolt by applying a force F, the torque Q is defined as
Q = r× F
1
9000-lb
75°
where r represents the position vector at whose tip the force F is applied. The direction of the torquevector indicates the axis and orientation of rotation. The magnitude of the torque is given by
|Q| = |F||r| sin θ.
Suppose that a bolt is tightened by applying a 80 N force to a 0.3 m wrench as shown. Find the magnitudeof the torque about the center of the bolt.
100
30° 60°
T1 T2
30° 30° T T
0.3 m θ = 75°
80 N
r
F
15°
v0
x
y
O α
d
λ
r
6. (Velocity and acceleration) A moving particle starts at an initial position r(0) = 〈1, 0, 0〉 with initialvelocity v(0) = i − j + k. Its acceleration is a(t) = 4t i + 6t j + k. Find its velocity and position atarbitrary time t. Assume all quantities are measured in respective units.
7. (Motion of a drone) The drone soars into the sky along the helical trajectory path r(t) = 〈4 cos t, 4 sin t, 4t〉,where t represents time.
(a) If the atmospheric velocity at arbitrary point is given by u = 〈−y, x, z/π〉, find the cosine of theangle between the atmospheric velocity and the velocity of the drone at t = π/2 (this angle is calledangle of attack in aeronautics).
(b) At t = 4π, the drone starts diving toward some point on the line segment between P (8, 0, 0) andQ(0, 8, 0). If the dive trajectory path is a line, find an equation of the plane that contains the linePQ and the dive path.
8. (Newton’s 2nd law) Consider a projectile having mass m is fired with angle of elevation α and initialvelocity v0. Assuming that aerodynamic drag is negligible and the external force is only due to gravity,the equation of motion is described by the Newton’s 2-nd law
md2r
dt2= −mg j.
(a) Integrating the equation twice in time, find the position function r(t) of the projectile.
100
30° 60°
T1 T2
30° 30° T T
0.25 m θ = 75°
40 N
r
F
15°
v0
x
y
O α
d
2
(b) Notice that the duration of the projectile being in air is the time until the y-position of the massbecomes equal to zero. Then the horizontal distance the projectile has traveled (= d: called range)is equal to the travel time times the x-component of the velocity ((dr/dt) · i). Find the range d.What value of α maximize the range d?
9. (Biology) The DNA molecule has the shape of a double helix as shown. The radius r of each helix isabout r = 10 angstroms (1 A = 10−8 cm). Each helix rises about λ = 34 A during each complete turn,and there are about N = 2.9× 108 complete turns.
100
30° 60°
T1 T2
30° 30° T T
0.25 m θ = 75°
40 N
r
F
15°
v0
x
y
O α
d
λ
r
(a) Verify the vector function of each helix can be parameterized as
r(t) =
⟨r cos t, r sin t,
λt
2π
⟩.
(b) Find the length L of each helix in terms of r, λ and N . Then compute the length L by plugging inthe given numbers.
10. (Elasticity) The elastic beam with a unit length is rigidly supported at one end O, and the vertical loadof strength W force unit is applied at the other end as shown. The vertical displacement y is given by
y =W
EI
(x2
2− x3
6
), 0 ≤ x ≤ 1,
where x is a coordinate along the beam, and E and I are mechanical parameters of the beam. Supposethat W/EI = 3/100 unit.
(a) Find curvature κ and local radius R of bending as functions of x.
(b) Considering that the slope of the bend is much less than unity (y′ 1), find approximated expres-sions of κ and R as functions of x. Also sketch the approximated κ(x) and R(x) along with thedisplacement y(x).
9000-lb
75°
45°
45°
T1
T2
T3
T4
A
B x
y
cable
W
R
y(x) x O
L = 1
Chapter 11 Partial Derivatives
1. (Thermal science) The thin metal plate located in the xy-plane has temperature T (x, y) at the point(x, y). The level curves of T are called isothermals because at all points on an isothermal the temperatureis the same. Suppose that the temperature distribution over the plate is given by
T (x, y) =100
1 + x2 + 2y2,
3
where T is measured in degree Celsius and x, y in meters.
(a) Sketch several isothermals.
(b) Find the rate change of temperature with respect to distance at the point (2, 1) in the x-direction.
(c) Find the rate change of temperature with respect to distance at the point (2, 1) in the y-direction.
2. (Electrical Engineering) The total resistance R produced by the electrical circuit having three conductorswith resistances R1, R2 and R3 shown below is given by the formula
1
R=
1
R1 +R2+
1
R3.
Find ∂R/∂R1.
R I
Roof: 5 units/m2day
East/West: 10 units/m2day
North/South: 8 units/m2day
Floor: 1 units/m2day
R1 R2
R3 R(t) I(t)
V(t)
3. (Ideal gas law) The gas law for a fixed mass m of an ideal gas at absolute temperature T , pressure P ,and volume V is given by PV = mRT , where R is the gas constant. Find the expression of
T
(∂P
∂T
)(∂V
∂T
).
4. (Ideal gas law) The pressure, volume, and temperature of a mole of an ideal gas are related by the idealgas law as given by PV = 8.31T , where P is measured in kilopascals, V in liters, and T in kelvins. Usedifferentials to find the approximate change in the pressure, if the volume increases from 12 L to 12.3 Land the temperature decreases from 310 K to 305 K.
5. (Electrical Engineering) If R is the total resistance of two resistors, connected in parallel, with resistancesR1 and R2, then
1
R=
1
R1+
1
R2.
If the resistance are measured in ohms as R1 = 20 Ω and R2 = 40 Ω, with a possible error 0.5 percent ineach case, estimate the maximum error in the calculated value of R.
R I
Roof: 5 units/m2day
East/West: 10 units/m2day
North/South: 8 units/m2day
Floor: 1 units/m2day
R1 R2
R3 R(t) I(t)
V(t)
R1 R2
6. (Electrical Engineering) The voltage V in a simple electrical circuit is slowly decreasing as the batterywears out. Also, the resistance R is slowly increasing as the resister heats up. Using the Ohm’s Law, V =IR, find how the current I is changing at the moment when R = 400 Ω, I = 0.08 A, dV/dt = −0.01 V/s,and dR/dt = 0.03 Ω/s.
R I
Roof: 5 units/m2day
East/West: 10 units/m2day
North/South: 8 units/m2day
Floor: 1 units/m2day
R1 R2
R3 R(t) I(t)
V(t)
4
7. (Astronautics) The lunar rover having mass m moves on the moon surface which topography is given byz = f(x, y), where (x, y) are plane coordinates and z is a surface elevation. Since the rover moves, thesecoordinates are functions of time t, i.e., x(t), y(t) and z(t). Suppose that the plane coordinates (x, y) aremeasured by an onboard inertial instrument and transmitted via telemetry to the mission control centerat NASA JPL at Pasadena, California. Suppose that the local topography data is available to NASAengineers from an earlier satellite mission and that the vehicle mass is nearly constant throughout themission. All length quantities are measured in meters and time is measured in seconds.
R I
(a) Find an expression of the velocity vector v of the vehicle in terms of available information at themission control center (x, y and f(x, y)).
(b) (Optional) Find an expression of the acceleration vector a as well.
(c) Suppose the vehicle is situated in a crater whose topography is given by z = (x2 + y2)/100, and thevehicle traverses a diverging spiral as given by x(t) = t cos(t/5), y(t) = t sin(t/5). Find the velocityvector as a function of time t.
8. (Electric potential) Suppose that over a certain region of space the electrical potential V is given byV (x, y, z) = 3y2 + 2xy + x2yz Volt where x, y and z are measured in meters.
(a) Find the rate of change of potential at P (1, 0, 2) in the direction toward the point (2, 1, 1).
(b) In which direction does V change most rapidly at P?
(c) What is the maximum rate of change of V at P?
9. (Civil Engineering) A rectangular building is being designed to minimize heat loss. The east and westwalls lose heat at a rate of 10 units/m2 per day, the north and south walls at a rate of 8 units/m2 perday, the floor at a rate of 1 unit/m2 per day, and the roof at a rate of 5 units/m2 per day. The volumemust be exactly 4000 m3. If there is no restriction in the wall height and lengths, find the dimensions ofthe building that minimize the heat loss.
R I
Roof: 5 units/m2day
East/West: 10 units/m2day
North/South: 8 units/m2day
Floor: 1 units/m2day
10. (Operating cost optimization) Suppose that the total weight of the airplane you are designing is 49, 000lbs that consists of fuel, payload (passengers) and airframe structure. For each flight, fuel costs 0.125 unittimes square of the fuel weight, operating labor costs 0.5 unit times square of the payload weight, andmaintenance service costs 0.25 unit times square of the structure weight. By using Lagrange multiplier,find weights of fuel, payload and structure that minimize the total operating cost per one flight.
11. Local temperature of a heated solid sphere bounded by x2 + y2 + z2 = 9 is given by f(x, y, z) =(x−1)2 +(y+2)2 +(z−2)2. Use the method of Lagrange multipliers to find the maximum and minimumvalues of the surface temperature of the sphere.
5
Chapter 12 Multiple Integrals
1. (Civil Engineering) Consider a utility water flow through a straight circular pipe with an inner radius ofR. Suppose the axial water velocity distribution depends on only radius r (distance from the pipe axis)as given by
U(r) = U0
[1−
( rR
)2],
where U0 (constant) is a velocity along the center line of the pipe.
x
y
𝜃𝜃 =𝜋𝜋4
𝜃𝜃 = −𝜋𝜋4
y
dV
x
y
z
x
z
O x
y dA D
x
y
O Ix
Iy
I0
r
R U(r)
(a) The volume flow rate Q is defined as an integral of the axial velocity over the flow section of thepipe (i.e., the volume flow rate is an amount of water flowing per unit time).
Q =
∫∫DU dA.
Find the volume flow rate.
(b) Find the average water velocity across the flow section (the average velocity is equal to the flowrate Q divided by the area of the cross section: vave = Q/πR2).
2. (Civil Engineering) The reservoir with full of still water of uniform density has a plane shape boundedby x = 3y2 and x = 3, measured x and y in unit length. The depth is given by the function h(x, y) =Hx(1 − y2), where the constant H is a maximum depth. Since the water has a constant density, thewater mass per unit plane area at any point (x, y) is equal to the density times the depth at the point.Denoting ρ as a water density per unit volume, calculate the total mass of water in terms of H and ρ.
3. (Electrical Engineering) If a electric charge is distributed over a plane region D and the charge density(in units of charge per unit area) is given by σ(x, y) at a plane point (x, y) in D, then the total chargeQ is given by
Q =
∫∫Dσ(x, y) dA.
Suppose that charge is distributed over the triangular region D bounded by y = 1− x, x = 1 and y = 1,and that the charge density at (x, y) is σ(x, y) = xy, measured in coulombs per square meter (C/m2).Calculate the total charge over D.
4. (Moment of Inertia 2D) The moment of inertia of a point mass m about an axis is defined as mr2 wherer is a distance from the mass to the axis. The moment of inertia is a measure of the resistance ofthe mass against a rotational motion about the axis. We can define the moment of inertia of a laminaby using double integrals. Denoting the local density function of the lamina as ρ(x, y), the moment ofinertia about x-axis (Ix) is defined as a double integral of the plane density ρ(x, y) times the square ofthe distance from (x, y) to x-axis:
Ix =
∫∫Dy2ρ(x, y) dA.
The moment of inertia about y-axis (Iy) is defined as a double integral of ρ(x, y) times the square of thedistance from (x, y) to y-axis:
Iy =
∫∫Dx2ρ(x, y) dA.
Then the moment of inertia about the origin, called polar moment of inertia, can be defined as a
6
x
y
𝜃𝜃 =𝜋𝜋4
𝜃𝜃 = −𝜋𝜋4
y x
z
h
L
y
dV
x
y
z
x
z
O x
y dA D
x
y
O Ix
Iy
I0
double integral of ρ(x, y) times the square of the distance from (x, y) to the origin:
I0 =
∫∫D
(x2 + y2)ρ(x, y) dA =
∫∫Dr2ρ(r, θ) dA = Ix + Iy.
Now, consider a ship propeller having four blades as shown. The plane shape of each blade is definedby the polar curve r = cos 2θ, −π/4 ≤ θ ≤ π/4. The polar moment of inertia of the propeller is fourtimes the polar moment of inertia of a blade. If the plane density is uniform over the blade denoted asρ(x, y) = ρ0, calculate the polar moment of inertia of the propeller. Assume all physical quantities aremeasured in respective units.
x
y
𝜃𝜃 =𝜋𝜋4
𝜃𝜃 = −𝜋𝜋4
5. (Earth Science) When studying the formation of mountain ranges, geologists estimate the amount ofenergy (work) required to lift a mountain from sea level (z = 0). Consider a mountain ridge having anuniform height of h above z = 0 and extending in y-direction by length L. Assume the ridge is boundedby planes z = h − x, z = h + x on sides and y = 0 and y = L at ends as shown. Suppose the weight
y x
z
h
L
density of the material is uniform, denoted as w. Assuming all the material is lifted from the sea level,the total work Q required in forming the mountain ridge is computed as a triple integral of the weightof dV in the ridge times its elevation from the sea level over the ridge region E, as given by
Q =
∫∫∫Ewz dV.
Calculate Q in terms of h, L and w.
6. Find the mass of the tetrahedron whose vertices are (0, 0, 0), (a, a, 0), (0, a, 0) and (0, 0, a), where a > 0.The volumetric mass density is given by the function ρ(x, y, z) = x/a. Express the center of mass astriple integrals as well. Do not evaluate the integrals.
7. (Moment of Inertia – Axisymetric geometry) The moment of inertia of the solid E about z-axis is definedas a triple integral of the mass of elemental volume dV times the square of the distance from dV to thez-axis, as given by
Iz =
∫∫∫E
(x2 + y2)ρ(x, y, z) dV.
7
x
y
𝜃𝜃 =𝜋𝜋4
𝜃𝜃 = −𝜋𝜋4
y x
z
h
L
y
dV
x
y
z
x
z
O
Now, consider the solid E that is bounded by the paraboloid z = x2 + y2 and the plane z = 1. Assumingthe density is uniform denoted as ρ(x, y, z) = K, find the moment of inertia Iz of the solid E. Assumeall physical quantities are measured in respective units.
8. (Moment of Inertia – Spherical geometry) Consider the solid sphere of radius a whose density at anypoint is the constant K times the distance from the center of the base. The center is positioned at theorigin O(0, 0, 0) and the base lies on the x-y plane. Find the moment of inertia of the solid about z-axis(Iz).
Chapter 13 Vector Calculus
1. (Electrostatics – Coulomb’s law) Suppose the electric charge Q is located at the origin. According tothe Coulomb’s law, the electric force F(r) exerted by this charge on another charge q located at point(x, y, z) with position vector r = 〈x, y, z〉 is
F(r) =εqQ
|r|3r,
where ε is called a Coulomb’s constant (ε = 8.988× 109 Nm2C−2). If qQ < 0, the force is attractive, andif qQ > 0, the force is repulsive.
I
B
x
y
z
−F
F
𝑄𝑄
𝑞𝑞
𝑥𝑥
𝑦𝑦
𝑧𝑧
Now, consider the particle with charge q moves along a straight line from (2, 0, 0) to (2, 1, 1) (measuredin meters) in the electric force field generated by the charge Q at the origin. Find the work done by theforce field to the charge q in terms of Q, q and ε.
2. (Electromagnetism – Ampere’s law) Experiments show that a steady current I in a long wire producesa magnetic field B that is tangent to any circle that lies in the plane perpendicular to the wire andwhose center is the axis of the wire as shown below. The Ampere’s law relates the electric current to itsmagnetic effects as given by ∫
CB · dr = µ0I
where I is the net current that passes through any surface bounded by a closed curve C and µ0 is aconstant called the permeability of free space.
(a) Take the path C to be a circle with radius r. By referring at the figure, explain why the magneticfield on the xy-plane can be expressed as
B = Bk×(r
r
), r = 〈x, y, 0〉 and r = |r|,
8
where B = |B| and (x, y) = (r cos t, r sin t) for 0 ≤ t ≤ 2π on the path C.
(b) Plugging in the magnetic field B into the Ampere’s law, and find the magnitude of the magneticfield (B = |B|) in terms of r, µ0 and I.
z
y
x
O k p
z0 r q
x
y r
C B
I
B
x
y
z
3. (Electrostatic potential) Consider the electric force field given by
F = εqQ
|r|3r
with the charge Q located at the origin and r = 〈x, y, z〉.
(a) Verify, whether the function
f(x, y, z) = − εqQ√x2 + y2 + z2
is a potential function of F.
(b) If the charge q moves from the point A to the point B whose distances from the origin are r1 andr2, respectively, find an expression of the work done by the electric force field.
4. (Polar moment of inertia – Green’s theorem) The lamina D has a shape bounded by the ellipse parame-terized as x = 2 cos t, y = sin t for 0 ≤ t ≤ 2π. If the mass density of the lamina is unity per unit area,the polar moment of inertia is defined by ∫∫
D(x2 + y2) dA.
If the functions P (x, y) and Q(x, y) satisfy the relation Qx − Py = x2 + y2, they can be chosen asP (x, y) = −y3/3 and Q(x, y) = x3/3. By applying the Green’s theorem, find the polar moment ofinertia. In calculation you can use the identities
sin4 t =1
8(3− 4 cos 2t− cos 4t) and cos4 t =
1
8(3 + 4 cos 2t+ cos 4t).
5. (Solid body rotation) Consider the solid body rotates about z-axis at a constant angular velocity ofω as shown. Orientation of the rotation follows the right-hand rule. Velocity at P (r = 〈x, y, z〉) is
I
B
x
y
z
−F
F
𝑄𝑄
𝑞𝑞
𝑥𝑥
𝑦𝑦
𝑧𝑧
v
z
y
x
O
P k
r θ
ω
9
proportional to ω and the distance from P to the z-axis, i.e.,
|v| = (|r| sin θ)ω.
Also, v is perpendicular to the z-axis (k) and−−→OP (r), and the direction follows the right-hand rule. This
implies that
v =k× r
|k× r||v| =
(k× r
|k||r| sin θ
)|v| =
(k× r
|r| sin θ
)(|r| sin θ)ω = ωk× r = ω × r,
where the vector ω = ωk defines an axis, direction and angular speed of rotation.
Now, calculate curl v. What is the direction and magnitude of curl v? (This gives you another interpre-tation of curl.)
6. (Conservative force field) The electrostatic force field of a point charge Q is given by
E =εQ
|r|3r.
(a) By taking curl of the field determine whether the field is conservative or nonconservative.
(b) For more general case in physics, if the inter-particle force field is given by F = f(r)r where r = |r|and f(r) is an arbitrary function, is F conservative or nonconservative?
7. (Center of mass) The nose cone of the missile has a shape of the paraboloid x = 2(y2 +z2) for 0 ≤ x ≤ 2.
(a) Parameterize the surface of the nose cone by using two independent parameters. State the domainof the parameters as well.
(b) If the nose cone is a shell having the mass density√
1 + 8x per unit area, find the total mass andcenter of mass of the nose cone.
8. (Electromagnetism – Gauss’ law) We know that The electric force field between two charges Q and qwith Q located at the origin is given by
F(r) =εqQ
|r|3r
Electric field E due to the charge Q is defined as the electric force per unit charge (E = F/q) as givenby
E =εQ
|r|3r.
In general the electric field E depends on distribution of charges (see the picture for example).
I
B
x
y
z
The Gauss’s law states that the net charge (sum of all charges) enclosed by a closed surface S is propor-tional to the surface integral of the electric field, as given by
Q = ε0
∫∫S
E · dS,
where the constant ε0 is called a permittivity of free space (ε0 ≈ 8.8542× 10−12 C2/N ·m2).
10
Now, in accordance with the Gauss law find the net charge contained in the solid sphere x2 + y2 + z2 ≤a2, if the electric field is given by E(x, y, z) = x i + y j + 2z k. (Hint: parameterize the sphericalsurface of radius a by using latitude angle (0 ≤ φ ≤ π) and polar angle (0 ≤ θ ≤ 2π), i.e., r =〈a sinφ cos θ, a sinφ sin θ, a cosφ〉. Then the normal vector is calculated as
rφ × rθ = (a sinφ) r = 〈a2 sin2 φ cos θ, a2 sin2 φ sin θ, a2 sinφ cosφ〉
which points outward the spherical domain.) Assume all physical quantities are measured in respectiveunits. Note that the evaluation of the surface integral becomes much easier, once you learn the divergencetheorem near the end of semester.
9. (Heat transfer – Fourier’s law) Suppose the temperature at a point (x, y, z) in a solid body is T (x, y, z).The Fourier’s law states that the heat flow is proportional to the temperature gradient as given by
F = −κ∇T
where κ is a constant called thermal conductivity of the substance. The sign is negative because heatflows from high temperature to low temperature region. The rate of heat flow across the surface S isgiven by the surface integral ∫∫
SF · dS = −κ
∫∫S∇T · dS.
We call this integral as a heat flux. The heat flux is a measure of the net thermal energy flowingacross the surface S per unit time. Now, suppose the solid alloy steel just taken out from the furnace
for heat treatment has a shape of the circular cylinder x2 + y2 = 4 bounded by planes z = 0 and z = 2,and the temperature distribution is given by T (x, y, z) = 980 − 2x2 − 2y2. By direct evaluation of thesurface integral, find the heat flux across the surface of the alloy steel in terms of κ. Assume all physicalquantities are measured in respective units. (Hint: parameterize the cylindrical surface by using polarangle (0 ≤ θ ≤ 2π) and z-coordinate as r = 〈2 cos θ, 2 sin θ, z〉)
10. (Electromagnetism – Faraday’s law) In electromagnetism, Faraday’s law states that if magnetic field Bchanges in time, then electric field E is induced, as expressed by the relation
curl E = −∂B
∂t,
where t represents time. Taking surface integral of this equation over the open surface S, we have∫∫S
curl E · dS = −∫∫
S
∂B
∂t· dS = − ∂
∂t
∫∫S
B · dS,
where we pulled the time derivative out of the integral. Here suppose that at certain time the electricfield is given by E = z2 i + 2xy j + 2xk and S is the hemisphere x2 + y2 + z2 = 1 for y ≥ 0 oriented inpositive y-axis. Find the time rate of change of the magnetic flux across S.
11. (Electromagnetism) In vacuum without electrical current, a surface integral of curl of the magnetic fieldB over an open surface S is equal to the time rate of change of flux of the electric field E across S dividedby a square of the speed of light c, as expressed by∫∫
Scurl B · dS =
1
c2∂
∂t
∫∫S
E · dS.
11
Suppose that the S is a cylindrical can with an open bottom, having cylindrical surface x2 + y2 = 1,0 ≤ z ≤ 1 and the top lid z = 1, x2 + y2 ≤ 1, oriented to positive z-axis. If the magnetic field at someinstant is given by B = (−y3e−z2 + x2e−x
2) i + (x3e−z
2+ y2e−y
2) j + e−z
2k, find the time rate of change
of flux of E across S in terms of c.
12. (Heat flux) Suppose the solid seamount (a mountain on ocean floor) has a shape of the cone z =4− 2
√x2 + y2 lying on the ocean floor z = 0 (x and y are in unit length). If the instantaneous heat flow
field inside and boundaries of the seamount is given by F = y(x+ sin y) i + z(cosx+ y) j + xz k physicalunit, calculate the flux of the heat flowing out of the conical surface of the seamount.
13. (Conservation of mass) Consider a fluid flow field with the local mass density ρ and the velocity u.The density times the velocity, ρu, gives an amount of fluid mass flowing per unit area and unit timeat a given point of the field. Consider an imaginary, non-deformable arbitrary solid region E with thebounding surface S in the flow field as shown.
The conservation of mass states that the flux of fluid flow across the boundary of E is equal to the lossof the fluid in E per unit time, as written by∫∫
Sρu · dS = − ∂
∂t
∫∫∫Eρ dV.
z
y
x
O k p
z0 r q
x
y r
C B
I
B
x
y
z
ρu dS
E S E
( mass of fluid flows out of S ) = ( loss of fluid mass in E )
(a) By applying the divergence theorem to the integral conservation relation, derive the differentialform of the conservation of mass, namely
div(ρu) = −∂ρ∂t.
(In derivation you can push ∂/∂t inside the triple integral and combine two integrals into one singleintegral. Note also, that E is an arbitrary volume.)
(b) If the flow field at some instant is given by ρu = xy i + (y3− 2yz) j + (z2− yz) k, find the time rateof change of the density at an arbitrary point of the field.
(c) If E is the sphere x2 + y2 + z2 = 1 and ρu is the field given in (b), find the time rate of change ofthe fluid mass of E.
12
