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Advanced Calculus, MATH–4600 Gregor Kovaˇciˇc Summer 2022
Homework Problems
- Plane Π 1 passes through the distinct points r 0 , r 1 , and r 2. Plane Π 2 passes through the distinct points r 0 , r 3 , and r 4. Write down the implicit equations of these planes. Do these planes intersect? If yes, write down the equation of their intersection. Discuss all possible cases.
- For the function z = f (x, y), use the chain rule to express the formula
x
∂z ∂y
− y
∂z ∂x
in terms of the polar coordinates x = r cos θ, y = r sin θ.
- Extra Credit: If f (0, 0) = 0 and
f (x, y) =
xy x^2 + y^2
, (x, y) 6 = (0, 0)
show that
∂f (x, y) ∂x
and
∂f (x, y) ∂y
exist at every point of R^2 , although f is not continuous at
the origin, (0, 0).
- Find the tangent plane and the normal line to the surface defined implicitly by the equation x^2 + y^2 − 8 z^2 = 0 at the point (2, − 2 , 1). (The normal line is the straight line perpendicular to the surface at the given point.) Attempt to find the corresponding two objects at the point (0, 0 , 0). What happens and why? Draw a picture and provide an explanation.
- Find the Taylor expansion around the origin of the function f (x, y, z) = sinh (xy + z^2 )
up to including terms of O
[
(x^2 + y^2 + z^2 )^3
]
. What order are the first neglected terms?
- Extra Credit: Define f (0, 0) = 0 and
f (x, y) =
xy (x^2 − y^2 ) x^2 + y^2
, (x, y) 6 = (0, 0).
(i) Show that |f (x, y)| < |xy| as x, y → 0, and so f is continuous at (0, 0). (ii) Calculate the expressions for fx(x, y) and fy(x, y) at (x, y) 6 = (0, 0), and also compute that the values of these partial derivatives at (x, y) = (0, 0) both equal 0. Then show that |fx(x, y)| < 2 |y| and |fy(x, y)| < 2 |x| and therefore fx and fy are continuous at (0, 0).
HINT: At an opportune moment, show and use the fact that |x^4 ± 4 x^2 y^2 − y^4 | ≤ 2 (x^2 + y^2 )^2.
(iii) Calculate the expressions for fxy(x, y) and fyx(x, y) at (x, y) 6 = (0, 0) and verify that they are the same. However, show that the limits of this expression along the x and y axes equal 1 and −1, respectively. (iv) From the definition, compute that fxy(0, 0) = −1 and fyx(0, 0) = 1.
- Determine how the location and type of the extrema of the function
f (x, y, z) =
x^2 − 2 αxy + xz − αy^2 + z^2
depend on the parameter α.
- Find the maximum of the function f (x, y, z) = x^2 y^2 z^2 on the sphere x^2 + y^2 + z^2 = c^2. Conclude the inequality ( x^2 y^2 z^2
)^13
x^2 + y^2 + z^2 3
which states that the geometric mean of three nonnegative numbers x^2 , y^2 , z^2 is never greater than their arithmetic mean.
- Extra Credit: Three points, P 1 , P 2 , and P 3 , with coordinates r 1 , r 2 , and r 3 , respectively, in the xy-plane, are the vertices of an acute-angled triangle. (Here rj = (xj , yj ), for j = 1, 2 , 3.) A fourth point P with coordinates r = (x, y) should be such that the sum of its distances dj from the points Pj , j = 1, 2 , 3, is the smallest. Characterize P in terms of the angles between the vector pairs r − ri and r − rj , i, j = 1, 2 , 3, i 6 = j. Then show that this P indeed gives the minimal distance sum.
HINT: To show that P is a minimum, show that the quadratic terms fxx(∆x)^2 +2fxy∆x∆y + fyy(∆y)^2 , where f = d 1 + d 2 + d 3 , can be expressed as a sum of squares.
- Consider the curve C parametrized by the expression
r(t) =
t +
a^2 t
, t −
a^2 t
, 2 a ln
t a
where a > 0 is a parameter. (i) Calculate the length of the curve C in the parameter interval a < t < b (ii) Calculate the Fren´et triad, curvature, and torsion for any value of the parameter t. (iii) Calculate the tangential and normal components of the acceleration for any value of the parameter t.
- (i) Show that a vector field of the form F(x, y, z) = ∇f (x, y, z), where the potential f (x, y, z) satisfies Laplace’s equation
∆f =
∂^2 f ∂x^2
∂^2 f ∂y^2
∂^2 f ∂z^2
is both incompressible and irrotational.
of C so that its projection on the xy plane is traversed counterclockwise with increasing
parameter. Then compute the integral
C
(2 − y) dx + (x + z − 1) dy + (2 − y) dz.
- Compute the area of the portion of the saddle-like surface z = bxy that lies inside the cylinder x^2 + y^2 ≤ a^2. (Draw a sketch!) What is the leading-order term in this area as either a → 0 or b → 0?
- Extra Credit: The surface S is the part of the sphere x^2 + y^2 + z^2 = a^2 that lies inside the cylinder x^4 + a^2 (y^2 − x^2 ) = 0. Sketch it and show that its area equals 2a^2 (π − 2).
HINT: Use cylindrical coordinates and take into account that S is symmetric with respect to both the xz- and yz-planes. Also, at some appropriate time(s) use the formula 1 + tan^2 θ = 1 cos^2 θ
- On a calm, rainy day, rain is falling straight down at the rate φv, where φ is its intensity per unit area, and v its velocity. Assume that your umbrella is the upper semicircle S : x^2 + y^2 + z^2 = a^2 , z > 0. Compute the volume flow rate that your umbrella deflects. Show also that, as far as this rate is concerned, it might as well be a disk with the same radius.
- Compute the surface integral I =
S
(∇ × F) · dS, where S is the portion of the ellipsoid
3 x^2 + 3y^2 + z^2 = 28 that lies above the plane z = 1, F = (yz^2 , 4 xz, x^2 y), and dS = n dS with n being the unit normal to S with a positive k component.
HINT: Use Stokes’ theorem.
- Compute the flow of the vector field F = r through the surface of the ellipsoid
x^2 a^2
y^2 b^2
z^2 c^2
= 1 in two different ways: (i) directly and (ii) using the divergence theorem.
HINT: Use appropriately modified spherical coordinates.
- Compute the integral
C
F·dr along the curve parametrized by r(t) = (t cos πt^3 , t sin πt^3 , t^4 ),
0 ≤ t ≤ 1, where F =
yz(2x + y + z), xz(x + 2y + z), xy(x + y + 2z)
- Find and sketch the curve y(x), which passes through the points (1, 0) and (2, −1), and
along which the functional J[y(x)] =
1
[
(y′) 2 − 2 xy
]
dx is extremal.
- Extra Credit: From Maxwell’s equations
∇ × E +
c
∂B
∂t
= 0, ∇ × H =
4 π c
J +
c
∂D
∂t
∇ · D = ρ, ∇ · B = 0,
show that the magnetic field B and the electric field E can be derived from a vector potential A and a scalar potential φ as
B = ∇ × A, E = −∇φ −
c
∂A
∂t
From these equations and the constitutive relations D = εE, B = μH, derive the forced wave equations for the vector and scalar potentials A and φ
∇^2 A −
εμ c^2
∂^2 A
∂t^2
4 πμ c
J = 0,
∇^2 φ −
εμ c^2
∂^2 φ ∂t^2
ρ ε
HINT: First, derive the last two equations with the right-hand sides ∇
∇ · A +
εμ c
∂φ ∂t
and −
∂t
∇ · A +
εμ c
∂φ ∂t
, respectively. Then argue that you can set ∇ · A +
εμ c
∂φ ∂t
because of the non-uniqueness of the vector potential A. Does this equation remove the non-uniqueness, and why?
- What two-dimensional geometric figure D with area A =
∂D
y dx − x dy has the
shortest perimeter?
HINT: Parametrize ∂D by r(t) = (x(t), y(t)), α < t < β, r(α) = r(β). The perimeter length
of D then equals
∫ (^) β
α
|r˙(t)| dt. Show that each of the two Euler’s equations represents a total
t-derivative, and consequently integrate it once. Manipulate the resulting two equations so that you find the usual implicit equation of a circle.
- (i) For Newton’s equation ¨x + x(1 − x) = 0, find the potential energy U (x), the kinetic energy, as well as the Lagrangian. What are the equilibrium points for this equation? (ii) Multiply Newton’s equation in part (i) by ˙x and integrate to obtain the total energy, E. Sketch U (x), and use it and E to sketch the trajectories of the equation in the x- ˙x-plane. (iii) Extra credit: What is the value of the energy along the separatrix loop?
- Find the geodesic curves on the unit sphere, that is, the shortest curves connecting pairs of points on it. What changes if the radius of the sphere is ρ?
HINT: Look for a function θ(φ). At some opportune point, rewrite sin φ
sin^2 φ − C 12 =
sin^2 φ
1 − C 12 (1 + cot^2 φ) (derive this!), and introduce a new variable u = cot φ. The final
result should yield z = ax + by on the sphere, where a and b are constants.
