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