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A comprehensive set of homework problems for an advanced calculus course, covering topics such as implicit equations of planes, chain rule in polar coordinates, partial derivatives, taylor expansions, extrema of functions, surface integrals, and geodesic curves. The problems are designed to challenge students' understanding of fundamental concepts and their ability to apply them in various contexts.
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Advanced Calculus, MATH–4600 Gregor Kovaˇciˇc Summer 2022
Homework Problems
x
∂z ∂y
− y
∂z ∂x
in terms of the polar coordinates x = r cos θ, y = r sin θ.
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).
up to including terms of O
(x^2 + y^2 + z^2 )^3
. What order are the first neglected terms?
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.
f (x, y, z) =
x^2 − 2 αxy + xz − αy^2 + z^2
depend on the parameter α.
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.
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.
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.
∆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.
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
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.
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.
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)
along which the functional J[y(x)] =
1
(y′) 2 − 2 xy
dx is extremal.
c
∂t
4 π c
c
∂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
∂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
εμ c^2
∂t^2
4 πμ c
∇^2 φ −
εμ c^2
∂^2 φ ∂t^2
ρ ε
HINT: First, derive the last two equations with the right-hand sides ∇
εμ c
∂φ ∂t
and −
∂t
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?
∂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.
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.