Calculus Answers

1. Write a complete Cayley Table for D6, the dihedral group of order 6.
2. Prove that if G is a group with property that the square of every element is the identity, then G is
abelian.
3. Construct the Cayley table for the group generated by g and h, where g and h satisfy the relations
g
3 = h
2 = e and gh = hg2
.
4. Let H and K be subgroups of a group G such that gcd(|H|, |K|) = 1. Apply Lagrange’s theorem to
show that |H ∩ K| = 1.
5. Consider the group Z12 and the subgroup H =< [4] >= {[0], [4], [8]}. Are the following pairs of elements
related under ∼H? Justify your answer.
(a) [3], [11],
(b) [3], [7],
(c) [5], [11],
(d) [6], [9],
(e) find all left cosets of H in G. Are they different from the right cosets?

1. Find the indicated derivatives:
a) u =
x+y
y+z
, x = p + r + t, y = p − r + t, z = p + r − t;
∂u
∂r .
b) y
5 + x
2
y
3 = 1 + yex
2
;
dy
dx .
c) ln(x + yz) = 1 + xy2
z
3
; ∂z/∂y
2. Let f(x, y) = x
2
y + x
3
y
2 and suppose you dont know what φ(t) = (x(t), y(t)) is, but you
know φ(2) = (1, 1), dx
dt (2) = 3, and dy
dt (2) = 1. Find the derivative of f(φ(t)) when t = 2.
3. Show that the following functions are functionally dependent and find a relation connect￾ing them:
f(x, y, z) = x + y + z, g(x, y, z) = x
2 + y
2 + z
2
, h(x, y, z) = xy + yz + xz
4. Find the local maxima, minima, and saddles of the functions h(x, y) = (2x−x
2
)(2y −y
2
).
5. Find the largest volume of a box with an open top, and surface area 100m2
.
6. Find the absolute minimum of f(x, y) = x
2 + y
2 + 2y − 1 on D = {(x, y)|x
2 +
y
2
4 ≤ 1}

Let O be the origin of coordinate plane, A, B lie on the upper half plane satisfying
OA=OB . If line OA has slope 1, line OB has slope −7 , what is the slope of line AB?

Calculate the work done by a force F vector = (x − y)i cap + xy j cap in moving a particle counter- clockwise along the circle x2 + y2 = 4 from the point (2, 0) to the point (0, −2

1. Compute the divergence and curl of each of the following vector fields:
a) hx
2 + y
2
, x2 − y
2
, z2
i
b) hx + y, x − y, zi
2. For any two vector fields F, G show that:
∇.(F × G) = G.(∇ × F) − F.(∇ × G).
3. Given scalar functions u(x, y, z), v(x, y, z), w(x, y, z), φ(x, y, z) and a vector field F(x, y, z) =
hu(x, y, z), v(x, y, z), w(x, y, z)i, prove that
div(φF) = φ(div F) + (∇φ) · F.
4. Find a function f such that F = ∇f where F = hyz, xz, xy + 2zi
5. Let r = hx, y, zi and r = |r|, verify that
a) ∇ · r = 3
b) div (r
n
r) = (n + 3)r
n
6. Given u = xy2
z
2 and v = yz − 3x
2
, find ∇ · [(∇u) × (∇v)]
7. Given the vector field F = he
x
sin y, ex
cos y, zi, show that F is irrotational, hence or
otherwise, find the functions f such that ∇f = F.

Obtain the directional derivative for a scalar field 2 3 2
φ(x, y,z) = 3x y − y z at the
point (1, −2, −1) in the direction .

Obtain first nd second derivative of f(x)=x^2sin y+ y^2cos x

obtain the fourier series for the following periodic function which has period of 2π:
f(x) = x^2 for -π≤x≤π

Obtain all the first second order partial derivatives of the function f(x,y) = x^2 siny + y^2 cosx

Given 2x^5+x^2-5/t^2 find day/dx