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# 数学代写|常微分方程代考Ordinary Differential Equations代写|Diferential Complexes and Grassman Algebras

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|Diferential Complexes and Grassman Algebras

First consider a scalar differential operator $P$ of order $m$, with complex-valued $C^{\infty}$ coefficients, in a $C^{\infty}$ manifold $\mathcal{M}$; “scalar” means that $P$ acts on $C^{\infty}$ functions and transforms them into $C^{\infty}$ functions. In a local chart $\left(\mathcal{U}, x_1, \ldots, x_n\right)$ we may represent $P$ as a differential operator
$$P(x, \mathrm{D})=\sum_{|\alpha| \leq m} c_\alpha(x) \mathrm{D}^\alpha$$
In the local chart $\left(\mathcal{U}, x_1, \ldots, x_n\right)$ its symbol is $P(x, \xi)$, its principal symbol $P_m(x, \xi)$ [see (2.1.8)]. Suppose we change coordinates: $x=x(y), \xi=\left(\frac{\partial y}{\partial x}\right)^{\top} \eta$ [see (9.3.5)]. The chain rule implies that the principal symbol of $P$ in the new coordinates is transformed into $P_m\left(x(y),\left(\frac{\partial y}{\partial x}\right)^{\top} \eta\right)$, which shows that $P_m$ can be regarded as a smooth function in the whole of the cotangent bundle $T^* \Omega$. In other words, $P_m(x, \xi)$ is coordinate free if the only permissible changes of coordinates in $T^* \Omega$ are those that leave the tautological one-form $\sigma$ unchanged. This is also true, trivially, of the zero-order term in $P$, equal to the action of $P$ on the function identically equal to 1 in $\mathcal{M}, P 1$. In general, however, this is not true of other parts of the symbol of $P$.

Example 9.4.1 Let $\mathcal{M}=\mathbb{K}, P=\frac{\mathrm{d}^2}{\mathrm{~d} x^2}$. Let $x \leadsto y$ be a regular change of variable in an open subset of $\mathbb{K}$. In the coordinate $y$ we have $P=\left(\frac{\mathrm{d} y}{\mathrm{dx}}\right)^2 \frac{\mathrm{d}^2}{\mathrm{~d} y^2}+\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2} \mathrm{~d} \mathrm{~d} y$. The homogeneous part of degree 1 of $P$ is modified.

Now let $\mathcal{B}_j(j=1,2)$ be two regular vector bundles over the same manifold $\mathcal{M}$ and let $\mathbb{C B}_j$ denote their complexifications (of course, $\mathbb{C} \mathcal{B}_j=\mathcal{B}_j$ when $\mathbb{K}=\mathbb{C}$ ). We introduce [cf. (9.2.3)] differential operators acting from sections over an open subset $\mathcal{U}$ of $\mathcal{M}$ of $\mathrm{CB}_1$ into those of $\mathbb{C B}_2$
$$P: C^{\infty}\left(\mathcal{U} ; \mathbb{C B}_1\right) \longrightarrow C^{\infty}\left(\mathcal{U} ; \mathbb{C B}_2\right)$$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Exterior algebra of a vector space, of a vector bundle

Let $\mathbf{E}$ be a vector space over $\mathbb{K}$ with $\operatorname{dim}_{\mathbb{K}} \mathbf{E}=N<+\infty$. We recall the definition of the exterior algebra of $\mathbf{E}$. For $p \geq 1$ we denote by $\mathbf{E}^{\otimes p}$ the $p^{\text {th }}$ tensor power of $\mathbf{E}$,
$$\mathbf{E}^{\otimes p}=\overbrace{\mathbf{E} \otimes \cdots \otimes \mathbf{E}}^{p \text { factors }} .$$

If $p \geq 2$ we denote by $\mathbf{N}^{(p)}$ the linear subspace of $\mathbf{E}^{\otimes p}$ spanned by the tensor products $\theta_1 \otimes \mathbf{x} \otimes \mathbf{x} \otimes \theta_2$ where $\mathbf{x} \in \mathbf{E}$ and $\theta_j \in \mathbf{E}^{\otimes p_j}, p_1+p_2=p-2$. For instance, in $\mathbf{E}^{\otimes 2}$ the two-tensor
$$\mathbf{x} \otimes \mathbf{y}+\mathbf{y} \otimes \mathbf{x}=(\mathbf{x}+\mathbf{y}) \otimes(x+y)-\mathbf{x} \otimes \mathbf{x}-\mathbf{y} \otimes \mathbf{y}$$
belongs to $\mathbf{N}^{(2)}$. We denote by $\Lambda^p \mathbf{E}$ the quotient vector space $\mathbf{E}^{\otimes p} / \mathbf{N}^{(p)}$; we also define $\Lambda^0 \mathbf{E}=\mathbb{K}$ and $\Lambda^1 \mathbf{E}=\mathbf{E}$.

We denote by $\mathbf{x}1 \wedge \cdots \wedge \mathbf{x}_p$ the coset in the quotient $\mathbf{E}^{\otimes p} / \mathbf{N}^{(p)}$ represented by $\mathbf{x}_1 \otimes \cdots \otimes \mathbf{x}_p \in \mathbf{E}^{\otimes p} ; \mathbf{x}_1 \wedge \cdots \wedge \mathbf{x}_p$ is often called a $p$-vector. We have $\mathbf{x}_1 \wedge \cdots \wedge \mathbf{x}_p=$ $\pm \mathbf{x}{i_1} \wedge \cdots \wedge \mathbf{x}_{i_p}$ if $\left(i_1, \ldots, i_p\right)$ is a permutation of $(1, \ldots, p)$, with the sign + or depending on whether the permutation is even or odd. By decomposing elements $\theta \in$ $\Lambda^p \mathbf{E}$ and $\omega \in \Lambda^q \mathbf{E}$ into sums of $p$-vectors $\mathbf{x}_1 \wedge \cdots \wedge \mathbf{x}_p$ and $q$-vectors $\mathbf{y}_1 \wedge \cdots \wedge \mathbf{y}_q$ we can define their exterior (or wedge) product $\theta \wedge \omega=(-1)^{p q} \omega \wedge \theta \in \Lambda^{p+q} \mathbf{E}$; obviously $(\theta, \omega) \longrightarrow \theta \wedge \omega$ is a bilinear map. When $p=0, \theta \wedge \omega$ is simply the product of the $q$-vector $\omega$ by the scalar $\theta$. It is checked directly that the exterior product is associative.

Select at random a linear basis of $\mathbf{E}, \mathbf{e}1, \ldots, \mathbf{e}_N$. It is readily seen that a basis of $\Lambda^p \mathbf{E}$ consists of the exterior products $\mathbf{e}_I=\mathbf{e}{i_1} \wedge \cdots \wedge \mathbf{e}_{i_p}$ where the multi-index $I=\left(i_1, \ldots, i_p\right)$ is ordered, in the sense that $1 \leq i_1<\cdotsN$ and that $\operatorname{dim} \Lambda^p \mathbf{E}=\left(\begin{array}{c}N \ p\end{array}\right)$ if $2 \leq p \leq N$.

# 常微分方程代写

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Diferential Complexes and Grassman Algebras

$$P(x, \mathrm{D})=\sum_{|\alpha| \leq m} c_\alpha(x) \mathrm{D}^\alpha$$

$$P: C^{\infty}\left(\mathcal{U} ; \mathbb{C B}_1\right) \longrightarrow C^{\infty}\left(\mathcal{U} ; \mathbb{C B}_2\right)$$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Exterior algebra of a vector space, of a vector bundle

$$\mathbf{E}^{\otimes p}=\overbrace{\mathbf{E} \otimes \cdots \otimes \mathbf{E}}^{p \text { factors }} .$$

$$\mathbf{x} \otimes \mathbf{y}+\mathbf{y} \otimes \mathbf{x}=(\mathbf{x}+\mathbf{y}) \otimes(x+y)-\mathbf{x} \otimes \mathbf{x}-\mathbf{y} \otimes \mathbf{y}$$

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