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# 数学代写|黎曼曲面代写Riemann surface代考|MA475 SIZES OF CELLS

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## 数学代写|黎曼曲面代写Riemann surface代考|SIZES OF CELLS

In this section we will discuss estimates for the sizes of pieces of chains like $F(K A)^r z$. We first develop some general theory of how to measure the size of a chain relative to a function, then apply it.

We should review conventions about wedge products. If $V$ is an inner product space, then $\Lambda^k V$ is given the norm which makes it an orthogonal quotient of $\otimes^k V$. Thus on a riemannian manifold, if $d x_1, \ldots, d x_k$ are an orthonormal set of cotangent vectors, then $\left|d x_1 \wedge \ldots \wedge d x_k\right|=1$. The volume element is a form of top degree of norm one; the integral of $d x_1 \wedge \ldots \wedge d x_k$ over a unit cube is 1 .

Let $M$ be a riemannian manifold, and suppose $\eta$ is a singular-de Rham chain of degree $k$ on $M$. It gives a linear functional $\int_\eta \phi$ of smooth $k$-forms $\phi$ (see §4), so $\eta$ can be considered as a current in the usual sense. If $\eta=h(u * H)$ for a piecewise polynomial map $h: H \rightarrow M$, then there is a constant $C$ such that $\sup \left|h^* \phi\right| \leq C \sup |\phi|$, so $\eta$ is continuous in the sup-norm. Therefore $\eta$ is representable by integration in the sense of Federer [8]. There is a positive measure $|\eta|$ such that
$$\left|\int_\eta \phi\right| \leq \int|\phi||\eta|,$$
and $|\eta|$ is minimal for this property $[8]$.

## 数学代写|黎曼曲面代写Riemann surface代考|Let r denote the coordinate on [0, 1]

Let $\tau$ denote the coordinate on $[0,1]$. The space of $(k+1)$-covectors on $M \times[0,1]$ decomposes as an orthogonal direct sum
$$\bigwedge^{k+1} T^(M \times[0,1])_{(m, r)}=\left(\bigwedge^k T^ M_m\right) \otimes(d \tau) \oplus \bigwedge^{k+1} T^* M_m$$
If $\phi$ is a $k+1$-form on $N$, we can write $f^(\phi)=\alpha_1 d r+\alpha_2$ according to the direct sum decomposition. Work at a point on $M \times[0,1]$, and at the corresponding image point in $N$. We may choose an orthonormal basis of covectors $d y_1, \ldots, d y_n$ in $N$ such that $\partial y_i / \partial \tau=0$ for $i \geq 2$. Thus $f^ d y_1=$ $\left(\partial y_1 / \partial \tau\right) d \tau+\beta_1$ with $\beta_1 \in T^* M$, and $f^* d y_i \in T^* M$ for $i \geq 2$. We can write $\phi=\phi_1 d y_1+\phi_2$ where $\phi_1$ and $\phi_2$ don’t involve $d y_1$. Then
$$f^* \phi=\left(f^* \phi_1\right)\left(\partial y_1 / \partial \tau\right) d \tau+\left(f^* \phi_1\right) \beta_1+f^* \phi_2 .$$
Thus $\alpha_1=\left(f^* \phi_1\right) \partial y_1 / \partial \tau$. Note that
$$\left|f^* \phi_1\right| \leq\left|\bigwedge^k \nabla f\right|_0|\phi|$$
and $\left|\partial y_1 / \partial \tau\right|=|\partial f / \partial \tau|$. Therefore
$$\left|\alpha_1\right| \leq\left|\bigwedge^k \nabla f\right|_0\left|\frac{\partial f}{\partial \tau}\right| f^*|\phi|$$

## 数学代写黎曼曲面代写Riemann surface代考|SIZES OF CELLS

$\sup \left|h^* \phi\right| \leq C \sup |\phi|$ ，所以 $\eta$ 在支持范数中是连续的。所以 $\eta$ 由费德勒 [8] 意义上的积分表示。有一个积极的措施 $|\eta|$ 这样
$$\left|\int_\eta \phi\right| \leq \int|\phi||\eta|,$$

## 数学代写|黎曼曲面代写Riemann surface代考|Let r denote the coordinate on [0, 1]

$$\left.\bigwedge^{k+1} T^{(} M \times[0,1]\right)_{(m, r)}=\left(\bigwedge^k T_m^M\right) \otimes(d \tau) \oplus \bigwedge^{k+1} T^* M_m$$

$$f^* \phi=\left(f^* \phi_1\right)\left(\partial y_1 / \partial \tau\right) d \tau+\left(f^* \phi_1\right) \beta_1+f^* \phi_2 .$$

$$\left|f^* \phi_1\right| \leq\left|\bigwedge^k \nabla f\right|_0|\phi|$$

$$\left|\alpha_1\right| \leq\left|\bigwedge^k \nabla f\right|_0\left|\frac{\partial f}{\partial \tau}\right| f^*|\phi|$$

## MATLAB代写

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