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# 物理代写|声学代写Acoustics代考|CDS389 The Product Rule or Integration by Parts

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## 物理代写|声学代写Acoustics代考|The Product Rule or Integration by Parts

If we have two functions of $x$, say $u(x)$ and $v(x)$, the product rule tells us how we can take the derivative of the product of those functions.
$$\frac{d(u v)}{d x}=\frac{d u}{d x} v+u \frac{d v}{d x}$$
This rule can be extended, for example, to the product of three functions of a single variable, where we have now added $w(x)$.

$$\frac{d(u v w)}{d x}=\frac{d u}{d x} v w+u \frac{d v}{d x} w+u v \frac{d w}{d x}$$
Although it is not the intent of this chapter to derive all the quoted results, it is instructive, in this context, to introduce the concept of a differential. ${ }^3$ If we say that a small change in $x$, by the amount $d x$, produces a small change in $u(x)$, by the amount $d u$, and that $|d u / u| \ll 1$, then we can show that Eq. (1.10) is correct by taking the product of the small changes.
$$d(u v)=(u+d u)(v+d v)-u v=u(d v)+v(d u)+(d u)(d v)$$
Since both $d u$ and $d v$ are small, the product of $d u$ times $d v$ must be much less than $u(d v)$ or $v(d u)$. If we make the changes small enough, then $(d u)(d v)$ can be neglected in Eq. (1.12). Throughout this textbook, we will make similar assumptions regarding our ability to ignore the products of two very small quantities [4].

The Fundamental Theorem of Calculus states that integration and differentiation are inverse processes. By rearranging Eq. (1.10) and integrating each term with respect to $x$, we can write an expression that allows for integration of the product of one function and the derivative of another function.
$$\int u(x) \frac{d v}{d x} d x=u(x) v(x)-\int v \frac{d u}{d x} d x$$
This result is known as the method of integration by parts.

## 物理代写|声学代写Acoustics代考|Logarithmic Differentiation

Acoustics and vibration are the “sciences of the subtle.” Most of our attention will be focused on small deviations from a state of stable equilibrium. For example, a sound pressure level ${ }^1$ of $115 \mathrm{~dB}_{\mathrm{SPL}}$ is capable of creating permanent damage to your hearing with less than $15 \mathrm{~min}$ of exposure per day [2]. That acoustic pressure level corresponds to a peak excess pressure of $p_1=16 \mathrm{~Pa}\left(1 \mathrm{~Pa}=1 \mathrm{~N} / \mathrm{m}^2\right)$. Since “standard” atmospheric pressure is $p_m=101,325 \mathrm{~Pa}$ [3], that level corresponds to a relative deviation from equilibrium that is less than 160 parts per million ( $\mathrm{ppm})$ or $p_1 / p_m=0.016 \%$.

If we assume that any parameter of interest (e.g., temperature, density, pressure) varies smoothly in time and space, we can approximate the parameter’s value at a point (in space or time) if we know the parameter’s value at some nearby point (typically, the state of stable equilibrium) and the value of its derivatives evaluated at that point. ${ }^2$ The previous statement obscures the true value of the Taylor series because it is frequently used to permit substitution of the value of the derivative, as we will see throughout this textbook.

Let us start by examining the graph of some arbitrary real function of position, $f(x)$, shown in Fig. 1.1. At position $x_o$, the function has a value, $f\left(x_o\right)$. At some nearby position, $x_o+d x$, the function will have some other value, $f\left(x_o+d x\right.$ ), where we will claim that $d x$ is a small distance without yet specifying what we mean by “small.”

The value of $f\left(x_o+d x\right)$ can be approximated if we know the first derivative of $f(x)$ evaluated at $x_o$.
$$f\left(x_o+d x\right) \cong f\left(x_o\right)+\left.\frac{d f}{d x}\right|_{x_0} d x$$
As can be seen in Fig. 1.1, the approximation of Eq. (1.1) produces a value that is slightly less than the actual value $f\left(x_o+d x\right)$ in this example. That is because the actual function has some curvature that happens to be upward in this case. The differential, $d x$, is used to represent both finite and infinitesimal quantities, depending upon context. For approximations, $\mathrm{d} x$ is assumed to be small but finite. For derivation of differential equations, it is assumed to be infinitesimal.

We can improve the approximation by adding another term to the Taylor series expansion of $f(x)$ that includes a correction proportional to the second derivative of $f(x)$, also evaluated at $x_o$. For the example in Fig. 1.1, the curvature is upward so the second derivative of $f(x)$, evaluated at $x_o$, is a positive number, so $\left(d^2 f / d x^2\right)_{x_o}>0$.

## 物理代写|声学代写Acoustics代考|积式法则或部件积分

$$\frac{d(u v)}{d x}=\frac{d u}{d x} v+u \frac{d v}{d x}$$

$$\frac{d(u v w)}{d x}=\frac{d u}{d x} v w+u \frac{d v}{d x} w+u v \frac{d w}{d x}$$虽然本章的目的不是推导所有引用的结果，但在此背景下，引入微分的概念是有指导意义的。 ${ }^3$ 如果我们说这是一个很小的变化 $x$，按数量计算 $d x$，产生一个小的变化 $u(x)$，按数量计算 $d u$，而且 $|d u / u| \ll 1$，则我们可以证明式(1.10)是正确的，取小变化的乘积。
$$d(u v)=(u+d u)(v+d v)-u v=u(d v)+v(d u)+(d u)(d v)$$

$$\int u(x) \frac{d v}{d x} d x=u(x) v(x)-\int v \frac{d u}{d x} d x$$

## 物理代写|声学代写Acoustics代考|对数分化

. 声学和振动是“微妙的科学”。我们的大部分注意力将集中在偏离稳定平衡状态的小偏差上。例如，声压等级${ }^1$或$115 \mathrm{~dB}_{\mathrm{SPL}}$，如果每天接触的声压低于$15 \mathrm{~min}$[2]，就会对你的听力造成永久性损伤。该声压级对应的峰值超压为$p_1=16 \mathrm{~Pa}\left(1 \mathrm{~Pa}=1 \mathrm{~N} / \mathrm{m}^2\right)$。由于“标准”大气压力为$p_m=101,325 \mathrm{~Pa}$[3]，该水平对应的相对偏离平衡小于百万分之160 ($\mathrm{ppm})$或$p_1 / p_m=0.016 \%$ .

. 0)

$$f\left(x_o+d x\right) \cong f\left(x_o\right)+\left.\frac{d f}{d x}\right|_{x_0} d x$$从图1.1中可以看出，Eq.(1.1)的近似产生的值略小于实际值 $f\left(x_o+d x\right)$ 在这个例子中。这是因为实际的函数有一个向上的曲率。微分， $d x$，用于表示有限和无穷小的量，具体取决于上下文。对于近似值， $\mathrm{d} x$ 假设是小而有限的。对于微分方程的推导，假设它是无穷小的 我们可以通过在$f(x)$的泰勒级数展开中加入另一项来改进近似，该项包含与$f(x)$的二阶导数成比例的修正，也在$x_o$处求值。对于图1.1中的例子，曲率是向上的，因此$f(x)$的二阶导数在$x_o$处的值是一个正数，因此$\left(d^2 f / d x^2\right)_{x_o}>0$。

## Matlab代写

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