Vector Analysis
MATH-UA 224 – VECTOR ANALYSIS
HOMEWORK 6 (DUE 4/28)
(1) Determine which of the following are tensors on R
4
, and express those that are in
terms of elementary tensors on R
4
:
(a) f(x, y) = 3x1y2 + 5x2x3
(b) f(x, y) = x1y2 + x2y4 + 1
(c) f(x, y, z) = 3x1x2z3 – x3y1z4
(d) f(x, y, z, u, v) = 5x3y2z3u4v4
(e) h(x, y, z) = x1y2z4 + 2x1z3
(2) Let f, g be the following tensors on R
4
f(x, y, z) =2x1y2z3 – x2y3z1
g =f2,1 – 5f3,1
(a) Express f ? g as a linear combination of elementary 5-tensors.
(b) Express (f ? g)(x, y, z, u, v) as a function.
(3) Which of the following are alternative tensors in R
4
?
(a) f(x, y) = x1y2 – x2y1 + x1y1
(b) g(x, y) = x1y3 – x3y1
(c) h(x, y) = (x1)
3
(y2)
3 – (x2)
3
(y1)
3
(4) Let s ? S5 be a permutation such that
(s(1), s(2), s(3), s(4), s(5)) = (3, 1, 4, 5, 2).
Show that s can be written as a composition of elementary permutations.
1

Vector Analysis
MATH-UA 224 – VECTOR ANALYSIS
HOMEWORK 6 (DUE 4/28)
(1) Determine which of the following are tensors on R
4
, and express those that are in
terms of elementary tensors on R
4
:
(a) f(x, y) = 3x1y2 + 5x2x3
(b) f(x, y) = x1y2 + x2y4 + 1
(c) f(x, y, z) = 3x1x2z3 – x3y1z4
(d) f(x, y, z, u, v) = 5x3y2z3u4v4
(e) h(x, y, z) = x1y2z4 + 2x1z3
(2) Let f, g be the following tensors on R
4
f(x, y, z) =2x1y2z3 – x2y3z1
g =f2,1 – 5f3,1
(a) Express f ? g as a linear combination of elementary 5-tensors.
(b) Express (f ? g)(x, y, z, u, v) as a function.
(3) Which of the following are alternative tensors in R
4
?
(a) f(x, y) = x1y2 – x2y1 + x1y1
(b) g(x, y) = x1y3 – x3y1
(c) h(x, y) = (x1)
3
(y2)
3 – (x2)
3
(y1)
3
(4) Let s ? S5 be a permutation such that
(s(1), s(2), s(3), s(4), s(5)) = (3, 1, 4, 5, 2).
Show that s can be written as a composition of elementary permutations.
1