Shifrin Math 3510 Day24: Differential forms

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Recommends Vivek Kaushik · 2017-07 · Mathematics Stack Exchange · on “What should I study to understand how to operate with differentials such as "dx"?”

You are asking about differential forms. Check out Barrett O'Neil's Differential Geometry, introduces forms quite nicely in the first chapter. I also recommend these lectures on Differential Forms [this video] by Professor Theodore Shifrin. Not only do his lectures deal with the setup of forms, but he proves the Generalized Stokes Theorem with them. Some background with linear algebra is helpful to fully understand the videos: Vector Space, Basis, Span, Linear Independence, Linear Transformations/Maps, Determinants, etc.

Recommends Antoni Parellada · 2018-04 · Mathematics Stack Exchange · on “Is the following summary of the wedge product correct?”

In trying to understand the exterior or wedge product [LINK] following Prof. Shifrin's series (Chapter 8 - here [this video] , here [LINK] and here [LINK] ), and connecting it with the more general tensor algebra [LINK] operations, I would like to make sure that the following is correct: The dual vector space $V^*=(\mathbb R^n)^*$ is a vector space of linear maps $\mathbb R^n \to \mathbb R$ that can be represented by a $1\times n$ matrix (i.e. row vector). It is isomorphic to $\mathbb R^n,$ and forms a vector space, which basis can be expressed as $\mathrm e^1,\mathrm e^2,\dots,\mathrm e^n,$ or as $\mathrm dx_1, \mathrm dx_2, \dots ,\mathrm dx_n,$ such that $\mathrm dx_i(\vec v)=\vec {\mathrm e}_

Cites JonnyPython · 2014-08 · Mathematics Stack Exchange · on “Differential forms and minor expansion, question about notation.”

There are lectures by Theodore Shifrin on differential forms, and sadly one video ends suddendly where he explains some notation. I try to formulate it in my own words: When k=n, we have $D(\vec{v_{1}},...,\vec{v_{n}})$ (where D means determinant). What about k=2, n=3 ? Now we have to take a random vector $\vec{a} \in R^{3}$. Take $\vec{a}= \vec{e_{1}}, \vec{e_{2}}, \vec{e_{3}}$. What do we get? $\begin{vmatrix} | & | & 1 \\ \vec{v_{1}} & \vec{v_{2}} & 0 \\ | & | & 0 \end{vmatrix} = \begin{vmatrix} v_{1,2} & v_{2,2} \\ v_{1,2} & v_{2,3} \end{vmatrix} $ Where, for example, $v_{1,2}$ means, take the second coordinate of $v_{1}$ and so on. Of course this is what you get when you do the cofactor

Cites Rahul Raphael Kanekar · 2017-10 · Mathematics Stack Exchange · on “Is $\mathrm{d}x_i$ a 0-form or a 1-form?”

I have just started studying differential forms and I understand that 0-forms are smooth real valued functions defined on $\mathbb{R}^n$, 1-forms are linear combinations of $\mathrm{d}x_i$ with smooth functions from $\mathbb{R}^n$ to $\mathbb{R}$ being coefficients, 2-forms are linear combinations of $\mathrm{d}x_i$∧$\mathrm{d}x_j$ with coefficients being smooth functions from $\mathbb{R}^n$ to $\mathbb{R}$ and so on. I understand that $\mathrm{d}x_i$, is the function that takes as input a vector in $\mathbb{R}^n$ and returns it's $\mathrm{i}^{th}$ component. So is $\mathrm{d}x_i$ a 0-form or a 1-form? Because by the definition of $\mathrm{d}x_i$, it is a smooth function from $\mathbb{R}^n$

Cites Clemens Bartholdy · 2021-06 · Mathematics Stack Exchange · on “What is a differential form?”

One day Élie Cartan was having a lot of Absinthe and suddenly he had a strange idea: "What if I take the change of variable formula and represent it without that annoying Jacobian determinant?". First Mr. Cartan expressed, as usual, the integrand and the region with new coordinate function $X' = \Phi(X)$ . Then he formally expressed the differentials with new differentials $$dx'= \sum \frac{\partial x'}{\partial x^*}dx$$ Now he could extract the Jacobian determinant if he treated differentials in a very bizarre way, $ dx \cdot dy = -dy \cdot dx$ , just like with matrix columns in determinant. Hence, he could get rid of the Jacobian determinant $\det [J]$ and encode that information into the

Cites Nate · 2023-05 · Mathematics Stack Exchange · on “Fundamental questions about differential forms:”

I have read through multiple [LINK] books [LINK] and watched Dr. Theodore Shifrin's video series [this video] . However, the questions that I had when going through much of the material have remained largely unanswered. They are: (1) Notational meaning: From my understanding, a differential form is a covector that takes as input a vector and returns some number. For instance take the 1-form $\omega = dx + 4dy$ . Is the "mapping" to some vector $v$ implicit in the definition? For instance, would we more rigorously be able to write out $\omega$ as $ \omega(v) = dx(v) + 4\cdot dy(v)$ ? Let $v$ = ( $\mathbf{i} + 2 \cdot \mathbf{j}$ ). Can we intuitively put the mapping as: $$ \omega(\mathbf{i} + 2 \c