Is the Following argument correct? Given a doubly indexed array $a_{mn}$ where $m,n\in\mathbf{N}$. Assume that $\lim_{m,n\to\infty}a_{mn} = a$, and given that for each fixed $m\in\mathbf{N}$, $\lim_{n\to\infty}(a_{mn}) = b_{m}$. Show that $\lim_{m\to\infty}b_m = a$. Proof. Let $\epsilon

(Wedge product of a 2-form with a 1-form).* Let $\omega$ be a $2-$form and $\tau$ a $1-$ form on $\mathbb R^3$. If $X, Y, Z$ are vector fields on $M$, find an explicit formula for $(\omega ∧\tau )(X,Y,Z)$ in terms of the values of $\omega$ and $\tau$ on the vector fields $X,Y,Z.$ Let ...

The exterior derivative of a $C^{\infty} 0-$ forms on an open set $U$ of $\mathbb R^n$ look like the total derivative of $f$ in the calculus. My doubt- Why did an exterior derivative of $C^{\infty} k-$ forms on an open set $U$ of $\mathbb R^n$ defined like this? What is the physical OR geom...

Let $V$ be a real vector space of dimension $n>2$ and $\omega\in \wedge^2V^*$ an alternating bilinear form on $V$. I'm wondering if there is a notion of exterior derivative $d:\wedge^2V^*\rightarrow \wedge^3V^*$ for alternating forms which is similar to the exterior derivative of differential fo...

I am studying exterior calculus, and I think I have some grasp of what is the exterior derivative. However its name still eludes me - why is it called a derivative? Is it just because the operator $d$ satisfies the Leibniz rule? I feel am looking for a way to grasp it as I understand most derivat...

Is there an example of a function $f:M\to \mathbb R$, where $M$ is a differentiable manifold, such that $f$ is constant on the hypersurface $\Sigma$ and its exterior derivative $df\neq 0$ on $\Sigma$? My intuition of the exterior derivative is simply how $f$ changes in any arbitrary direction. ...

I know there is an intimate relation between covariant, Lie and exterior derivative. I know that the covariant derivative requires more structure than the exterior, so it would be possible. How do I express a covariant derivative $\nabla_{X} Y$ in terms of the exterior derivative, assuming the Le...