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Advanced Calculus Homework Problems: MATH-4600, Assignments of Mathematics

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.

Typology: Assignments

2021/2022

Uploaded on 12/12/2024

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Advanced Calculus, MATH–4600 Gregor Kovaˇciˇc
Summer 2022
Homework Problems
1. Plane Π1passes through the distinct points r0,r1, and r2. Plane Π2passes through the
distinct points r0,r3, and r4. 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.
2. For the function z=f(x, y), use the chain rule to express the formula
x∂z
∂y yz
∂x
in terms of the polar coordinates x=rcos θ,y=rsin θ.
3. Extra Credit: If f(0,0) = 0 and
f(x, y) = xy
x2+y2,(x, y)6= (0,0)
show that f (x, y)
∂x and f(x, y)
∂y exist at every point of R2, although fis not continuous at
the origin, (0,0).
4. Find the tangent plane and the normal line to the surface defined implicitly by the
equation x2+y28z2= 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.
5. Find the Taylor expansion around the origin of the function f(x, y, z ) = sinh (xy +z2)
up to including terms of Oh(x2+y2+z2)3i. What order are the first neglected terms?
6. Extra Credit: Define f(0,0) = 0 and
f(x, y) = xy (x2y2)
x2+y2,(x, y)6= (0,0).
(i) Show that |f(x, y)|<|xy|as x, y 0, and so fis 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 fxand fyare continuous at (0,0).
HINT: At an opportune moment, show and use the fact that |x4±4x2y2y4| 2 (x2+y2)2.
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Advanced Calculus, MATH–4600 Gregor Kovaˇciˇc Summer 2022

Homework Problems

  1. 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.
  2. 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.

  1. 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).

  1. 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.
  2. 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?

  1. 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.

  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.

  1. 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.

  1. 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.

  1. 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.

  1. (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.

  1. 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?
  2. 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

  1. 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.
  2. 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.

  1. 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.

  1. 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)

  1. 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.

  1. 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?

  1. 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.

  1. (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?
  2. 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.