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$M$ looks like this: $M(u) = (F_{12}(u),\dots,F_{1n}(u),F_{23}(u),\dots F_{2n}(u)),\dots ,F_{nn}(u))$ $M(u) = (F_{12}(u),F_{21}(u),\dots) = (F_{12}(u),-F_{12}(u),\dots)$ doesn't make much sense. then the first variation: $\delta M(u) = \frac{d}{d\epsilon} M(F_{12}(u+\epsilon v),\dots,F_{nn}(u+\epsilon v))\big|_{\epsilon = 0}$ using the chain rule: ...

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The equation you wish to prove is linear in each of the vectors $u_j$ and $v_j$. Therefore it suffices to show the identity when these vectors are basis vectors. There are $n$ basis vectors from which we now want to choose the $n$ vectors $u_1,\dots,u_n$. If we choose any two to be the same, then both sides of the identity vanish (and the identity is true), ...

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Hint : (1) Let $$U=[u_1\cdots u_n],\ V=[v_1\cdots v_n]$$ Then $$(U^TV)_{ij} = u_i\cdot v_j =g(u_i,v_j)$$ (2) ${\rm det} (U^TV)={\rm det}\ U {\rm det}\ V$

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Given a $p$-form $\theta \in \bigwedge^p E^*$, we can define an alternating multilinear map $h \colon E^{n-p} \to \bigwedge^n E$ by $$h(u_1, \ldots, u_{n-p}) = \theta \wedge \tilde{u}_1 \wedge \ldots \wedge \tilde{u}_{n-p}.$$ Let $b \colon \mathbb{R} \to \bigwedge^n E$ be the linear map $$b(t) = t \omega.$$ Because $\bigwedge^n E$ is one-dimensional and ...

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As by the question linked by Michael, it is generally proved in a course about differential forms that $\alpha\wedge\beta = (-1)^{pq} \beta\wedge\alpha$ where $p$ and $q$ are the degrees of $\alpha$, $\beta$. Using this the result is immediate. But you could try and prove it, at least in your specific case. Hint: it should not be difficult to prove it for ...

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Try to formalize what internal stress state should actually mean. What science has come up with is the notion of sectional forces and moments, together with a way to relate them to internal stress state. You need to do a Gedankenexperiment: Imagine a smooth cut passing through the material, including the point $p$. Remove the material on one side of the cut ...

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To remove notational confusion, let $\lambda^1,\ldots,\lambda^n$ be the dual basis of $\frac{\partial}{\partial x^1},\ldots,\frac{\partial}{\partial x^n}$. Then, your goal is to show that $$d(x^i)=\lambda^i.$$ i.e. $d(x^i)=dx^i$, where the later is the formal symbol usually used for $\lambda^i$. Note that this justifies the notation $dx^i$. Now, compute: ...

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Let's start from the top. If you have any programming background, then the following might make some more sense to you: A tensor is merely a function that is linear on each of its arguments and produces a number. For instance, let $f: \mathbb R^n \to \mathbb R$. That is, for any vector $v \in \mathbb R^n$, $f(v)$ is some real number. If $f$ is ...

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Take a look at Couple stress theory for solids

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The oriented volume spanned by three vectors is a multilinear alternating map of the vectors themselves. That's the starting point. See, e.g., the book by S.Winitzki.

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