Ryan Vaughn

The reading group maintains a set of notes and solutions to exercises. The PDF copies will be periodically updated to this page. Currently, the notes are written and maintained by Cigole Thomas (Thank you!).

The last upload was made on 03/03/2018.

DG Notes

DGExercises

Chapter 1:
Book Problems:

1.7.5 – Me
1.7.6

Extra Problems:

Let $p\in \mathbb{R}^n$ be fixed. Show that the tangent space $T_p \mathbb{R}^n = \{ v_p = (v,p) : v\in \mathbb{R}^n\}$ is linearly isomorphic to the set $\mathcal{V}_p = \{v_p:C^\infty (\mathbb{R})\rightarrow \mathbb{R}: v_p \textnormal{ is linear, } v_p(fg)=f(p)v_p(g)+g(p)v_p(f)\}$ .
– Me

Chapter 2:

Book Problems:

2.6.1
2.6.2
2.6.3
Prove Corollary 2.5.4

Extra Problems:

When introducing the concept of vector fields along a curve in Section 2.2, O’Neil asserts that taking the derivative of a curve $\alpha$ is a vector field along $\alpha$ . He then asserts that one can obtain another vector field along $\alpha$ by taking the second derivative of $\alpha$ . Can this be expressed via the covariant derivative? If so, how? (Harold has done this, but the solution should be added to the $\LaTeX$ document.
Prove that the area of a $n$ -dimensional parallelopiped spanned by $n$ vectors $v_1,...,v_n$ is given by the determinant of the matrix whose columns are $(v_1,..,v_n)$ .