1617-257/Homework Assignment 14

Doing

Solve all the problems in sections 27-28, but submit only your solutions of problem 1 in section 27 and problems 1 and 6 in section 28. In addition, solve the following problems three problems, though submit only your solutions of the first two. These problems require more creativity than the usual and the are less precisely specified. You may wish to listen carefully at the tutorials this week!

Problem A. Along the lines of our development of a theory of "tensors" and a theory of "alternating tensors", develop a theory of "symmetric tensors" ${\displaystyle S^{k}(V)}$ (a symmetric tensor is a tensor whose values are unchanged if its arguments are permuted). Your theory should include definitions for specific tensors ${\displaystyle \sigma _{I}}$ for ${\displaystyle I\in {\underline {n}}_{s}^{k}}$ (what should ${\displaystyle {\underline {n}}_{s}^{k}}$ be)?), a proof that the ${\displaystyle \sigma _{I}}$ exist and are unique and that they form a basis of ${\displaystyle S^{k}(V)}$, and a computation of the dimension of ${\displaystyle S^{k}(V)}$.

Problem B. Find a good way of identifying ${\displaystyle A^{1}({\mathbb {R} }^{3})}$ and ${\displaystyle A^{2}({\mathbb {R} }^{3})}$ with ${\displaystyle {\mathbb {R} }^{3}}$. Under that identification, ${\displaystyle \wedge \colon A^{1}({\mathbb {R} }^{3})\times A^{1}({\mathbb {R} }^{3})\to A^{2}({\mathbb {R} }^{3})}$ becomes a map ${\displaystyle P\colon {\mathbb {R} }^{3}\times {\mathbb {R} }^{3}\to {\mathbb {R} }^{3}}$. If you chose your identifications right, ${\displaystyle P}$ is the vector product of two vectors in ${\displaystyle {\mathbb {R} }^{3}}$. See to it that this is the indeed the case!

Problem C. The determinant, as a function of a list of column vectors, is alternating. Write it in terms of the elementary alternating functions ${\displaystyle \psi _{I}}$.

Submission

Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

This assignment is due in class on Wednesday February 15 by 2:10PM.

Important

Please write on your assignment the day of the tutorial when you'd like to pick it up once it is marked (Wednesday or Thursday).