Ryan Vaughn

Meeting 1: Tangent Vectors and Directional Derivatives (02/01/2018)

Summary: Introduced tangent spaces of Euclidean space, as well as partial and directional derivatives. If we fix a tangent vector $v_p \in T_p \mathbb{R}^n$ , we can think of the directional derivative as a mapping $v_p: C^\infty(\mathbb{R}^n)\rightarrow \mathbb{R}$ defined by mapping a function $f$ to it’s directional derivative in the direction $v_p$ (denoted $v_p[f]$ .) When studying abstract smooth manifolds, many authors define tangent vectors as mappings from $C^\infty(M) \rightarrow \mathbb{R}$ which are linear and satisfy the product rule. The directional derivative is where this abstract definition comes from. Introduced vector fields, the Euclidean standard frame fields, and defined smoothness of vector fields.

Corresponding Textbook Sections: O’Neil: Sections 1.1-1.3

Meeting 2: Curves in $\mathbb{R}^n$ and Mappings (02/07/2018)

Summary: Curves are smooth functions from $(a,b)\rightarrow \mathbb{R}^n$ . For every tangent vector $v_p$ , we can define a unique straight-line curve $\alpha(t) = p +tv$ . Every smooth function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ induces a linear map $f_*|_p : T_p\mathbb{R}^n \rightarrow T_{f(p)}\mathbb{R}^m$ . This map is defined by assigning $v_p$ to the unique straight-line curve through $p$ in the direction $v$ , then applying $f$ to the curve. The resulting tangent vector obtained by differentiating the resulting curve at the point $t=0$ is the value of $f_*(v_p)$ . The matrix representation of $f_*|_p$ in terms of the standard coordinate bases is the Jacobian matrix from multivariable calculus.

Corresponding Textbook Sections: O’Neil: Sections 1.4 and 1.7.

Meeting 3: Problem Discussion (02/14/2018)

Summary: We presented several problems from O’Neil section 1.7.

Corresponding Textbook Sections: O’Neil: Section 1.7

Meeting 4: The Covariant Derivative (02/21/2018)

Summary: We introduced the covariant derivative in $\mathbb{R}^n$ . The covariant derivative $\nabla_W V$ is a way of differentiating a vector field $V$ in the direction of another vector field $W$ . It is computed by the following steps:

Write the vector field field $V$ in terms of the standard coordinate frame in $\mathbb{R}^n$ as $V_p = \sum_{i=1}^n v_i(p) U_i(p)$ .
Gather the real valued functions $v_1,...,v_n$ into a function $F:\mathbb{R}^n\rightarrow \mathbb{R}^n$ by $F(p)=(v_1(p),...,v_n(p)$ .
Compute $F_*$ .
For each $p \in \mathbb{R}^n$ , define the vector field $(\nabla_W V)_p$ by $(\nabla_W V)_p = F_*(W_p).$

The above steps give us a mapping $\nabla$ which takes in pairs of vector fields $(V,W)$ and outputs a vector field. Notice that we are taking the output $F_*(W_p)$ and considering it as a vector based at $p$ , when technically it should be a vector based at $F(p)$ .

Corresponding Textbook Sections: O’Neil: Section 2.5

Meeting 5: Differential 1-forms (02/28/2018)
A covector is an element of the dual space of $T_p(\mathbb{R}^n)$ , the set of covectors on $T_p(\mathbb{R}^n)$ is denoted $T_p^*(\mathbb{R}^n)$ . In our case, covectors can be thought of as a way to measure the “signed, 1-dimensional area” of a vector. For each $C^\infty(\mathbb{R}^n)$ function $f$ , one can construct a covector $df|_p$ defined by mapping a tangent vector $v \in T_p(\mathbb{R}^n)$ to the directional derivative of $f$ in the direction $v$ , denoted $v[f]$ . Taking the projection functions $x^i:\mathbb{R}^n \rightarrow \mathbb{R}$ , the set $(dx^i)_{i=1}^n$ is a basis for $T^*_p(\mathbb{R}^n)$ .

Corresponding Textbook Sections: O’Neil: Section 1.5 (Content was also taken from John Lee’s Introduction to Smooth Manifolds)

Meeting 6: Differential k-forms (03/07/2018)
Defined differential 1-forms as covector fields on $\mathbb{R}^n$ . Gave a very informal introduction to $k$ -forms as alternating $k$ -tensor fields. A (covariant) $k$ -tensor is a $k$ -multilinear map on $T_p\mathbb{R}^n$ . We showed that an alternating $k$ -tensor is the ideal map to measure signed $k$ -dimensional area of a parallelopiped spanned by $k$ linearly independent vectors in $T_p\mathbb{R}^n$ .

Corresponding Textbook Sections: Various sections of Lee’s Introduction to Smooth Manifolds.

Meeting 7: Differential k-forms, Part 2 (03/21/2018)
Discussed how to construct high rank tensors from lower rank tensors. This can be done in a variety of ways. First, we discussed the tensor product $\otimes$ operation. The tensor product of a $k$ tensor $\alpha$ and an $\ell$ tensor $\beta$ is the tensor $\alpha \otimes \beta$ constructed by simply multiplying the outputs of $\alpha$ and $\latex \beta$ together. Unfortunately, the tensor product is insufficient for our purposes, since it does not preserve the alternating property. If $\alpha$ and $\beta$ are both alternating tensors, it is possible that $\alpha \otimes \beta$ is not alternating. We thus introduced the wedge product, which is like the tensor product, but preserves the alternating property.

Meeting 8: The Exterior Derivative (03/21/2018)
We discussed differential $k$ forms and how to compute the exterior derivative of a differential $k$ form in $\mathbb{R}^n$ . The exterior derivative $d$ is a mapping from the set of $k$ forms to the set of $k+1$ forms. It is linear with respect to $\mathbb{R}$ and represents the most general form of the derivative. In order to compute the exterior derivative, we construct a basis for the set of $k$ forms.

Meeting 9: Problems Session (03/21/2018)

Meeting 10: De Rham Cohomology (03/21/2018)