# 数学 Maths 2 PHYS130001/PHYS238001

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Let $b_{\alpha \beta \mid \sigma}:=\partial_{\sigma} b_{\alpha \beta}-\Gamma_{\alpha \sigma}^{\tau} b_{\tau \beta}-\Gamma_{\beta \sigma}^{\tau} b_{\alpha \tau}$ denote the first-order covariant derivatives of the curvature tensor, defined here by means of its covariant components. Show that these covariant derivatives satisfy the Codazzi-Mainardi identities
$$b_{\alpha \beta \mid \sigma}=b_{\alpha \sigma \mid \beta}$$
which are themselves equivalent to the relations (Thm. 2.8-1)
$$\partial_{\sigma} b_{\alpha \beta}-\partial_{\beta} b_{\alpha \sigma}+\Gamma_{\alpha \beta}^{\tau} b_{\tau \sigma}-\Gamma_{\alpha \sigma}^{\tau} b_{\tau \beta}=0$$
Hint: The proof is analogous to that given in for establishing the relations $\left.b_{\beta}^{\tau}\right|{\alpha}=\left.b{\alpha}^{\tau}\right|_{\beta}$.

## PPHYS130001/PHYS238001COURSE NOTES ：

$u_{i}^{\varepsilon}\left(x^{\varepsilon}\right)=u_{i}(\varepsilon)(x)$ for all $x^{\varepsilon}=\pi^{\varepsilon} x \in \bar{\Omega}^{\varepsilon}$,
where $\pi^{c}\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}, x_{2}, \varepsilon x_{3}\right)$. We then assume that there exist constants $\lambda>0, \mu>0$ and functions $f^{i}$ independent of $\varepsilon$ such that
$$\begin{gathered} \lambda^{\varepsilon}=\lambda \text { and } \mu^{\varepsilon}=\mu, \ f^{i, \varepsilon}\left(x^{\varepsilon}\right)=\varepsilon^{p} f^{i}(x) \text { for all } x^{\varepsilon}=\pi^{e} x \in \Omega^{\varepsilon}, \end{gathered}$$

# 数学 MATH MATHS2025_1/MATHS3016_1

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Let $f$ be a function from a nonempty open subset $E$ of $\mathbb{R}$ to $\mathbb{R}$. The function $f$ is said to be differentiable at $c \in E$ if
$$\lim {x \rightarrow c} \frac{f(x)-f(c)}{x-c}$$ or, equivalently, $$\lim {h \rightarrow 0} \frac{f(c+h)-f(c)}{h}$$
exists. This limit (if it exists) is called the derivative of $f$ at $c$. If the derivative of $f$ exists at every $c \in E$, then $f$ is said to be differentiable on $E$ (or just differentiable). The derivative of $f$ as a function from $E$ to $\mathbb{R}$ is denoted by
$$f^{\prime} \text { or } \frac{d f}{d x}$$
Note that the limit in Eq. (7.1) is understood as the limit of the function
$$g(x)=\frac{f(x)-f(c)}{x-c}, \quad x \in E \backslash{c}$$

## MATHS2025_1/MATHS3016_1COURSE NOTES ：

From the definition of $f^{\prime}(c)$, it follows that for every $\varepsilon>0$, there exists $\delta>0$ such that $x \in E,|x-c|<\delta$, and $x \neq c$ imply
$$\left|\frac{f(x)-f(c)}{x-c}-f^{\prime}(c)\right|<\varepsilon .$$
Thus, for every $x \in E$ with $|x-c|<\delta$,
$$|f(x)-\varphi(x)| \leq \varepsilon|x-c|,$$
where $\varphi$ is the linear function defined by
$$\varphi(x)=f(c)+f^{\prime}(c)(x-c), \quad x \in \mathbb{R}$$