# 数学代写|应用数学代写applied mathematics代考|Math2090

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• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
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## 数学代写|应用数学代写applied mathematics代考|The Euler-Lagrange equation

In the mechanical problems considered above, the Lagrangian is a quadratic function of the velocity. Here, we consider Lagrangians with a more general dependence on the derivative.
Let $\mathcal{F}$ be a functional of scalar-valued functions $u:[a, b] \rightarrow \mathbb{R}$ of the form
\begin{aligned} & \mathcal{F}(u)=\int_a^b F\left(x, u(x), u^{\prime}(x)\right) d x \ & F:[a, b] \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \end{aligned}
where $F$ is a smooth function.
It is convenient to use the same notation for the variables
$$\left(x, u, u^{\prime}\right) \in[a, b] \times \mathbb{R} \times \mathbb{R}$$
on which $F$ depends and the functions $u(x), u^{\prime}(x)$. We denote the partial derivatives of $F\left(x, u, u^{\prime}\right)$ by
$$F_x=\left.\frac{\partial F}{\partial x}\right|{u, u^{\prime}}, \quad F_u=\left.\frac{\partial F}{\partial u}\right|{x, u^{\prime}}, \quad F_{u^{\prime}}=\left.\frac{\partial F}{\partial u^{\prime}}\right|_{x, u}$$

If $h:[a, b] \rightarrow \mathbb{R}$ is a smooth function that vanishes at $x=a, b$, then
\begin{aligned} d \mathcal{F}(\vec{u}) h & =\left.\frac{d}{d \varepsilon} \int_a^b F\left(x, u(x)+\varepsilon h(x), u^{\prime}(x)+\varepsilon h^{\prime}(x)\right) d x\right|{\varepsilon=0} \ & =\int_a^b\left{F_u\left(x, u(x), u^{\prime}(x)\right) h(x)+F{u^{\prime}}\left(x, u(x), u^{\prime}(x)\right) h^{\prime}(x)\right} d x . \end{aligned}

## 数学代写|应用数学代写applied mathematics代考|Derivation of Newton’s resistance functional

Following Newton, let us imagine that the gas is composed of uniformly distributed, non-interacting particles that reflect elastically off the body. We suppose that the particles have number-density $n$, mass $m$, and constant velocity $v$ the downward $z$-direction, in a frame of reference moving with the body.

We assume that the body is cylindrically symmetric with a maximum radius of $a$ and height $h$. We write the equation of the body surface in cylindrical polar coordinates as $z=u(r)$, where $0 \leq r \leq a$ and
$$u(0)=h, \quad u(a)=0$$
Let $\theta(r)$ denote the angle of the tangent line to the $r$-axis of this curve at the point $(r, u(r))$. Since the angle of reflection of a particle off the body is equal to the angle of incidence, $\pi / 2-\theta$, the reflected particle path makes an angle $2 \theta$ to the $z$-axis.

The change in momentum of the particle in the $z$-direction when it reflects off the body is therefore
$$m v(1+\cos 2 \theta)$$
For example, this is equal to $2 m v$ for normal incidence $(\theta=0)$, and 0 for grazing incidence $(\theta=\pi / 2)$.

The number of particles per unit time, per unit distance in the radial direction that hit the body is equal to
$2 \pi n v r$
Note that $[2 \pi n v r]=\left(1 / L^3\right) \cdot(L / T) \cdot(L)=1 /(L T)$ as it should.
The rate at which the particles transfer momentum to the body per unit time, which is equal to force $F$ exerted by the gas on the body, is given by
$$F=2 \pi n m v^2 \int_0^a r(1+\cos 2 \theta) d r$$

# 应用数学代考

## 数学代写|应用数学代写applied mathematics代考|The Euler-Lagrange equation

$$\mathcal{F}(u)=\int_a^b F\left(x, u(x), u^{\prime}(x)\right) d x \quad F:[a, b] \times \mathbb{R}$$

$$\left(x, u, u^{\prime}\right) \in[a, b] \times \mathbb{R} \times \mathbb{R}$$

$$F_x=\frac{\partial F}{\partial x}\left|u, u^{\prime}, \quad F_u=\frac{\partial F}{\partial u}\right| x, u^{\prime}, \quad F_{u^{\prime}}=\left.\frac{\partial F}{\partial u^{\prime}}\right|_{x, u}$$

## 数学代写|应用数学代写applied mathematics代考|Derivation of Newton’s resistance functional

$$u(0)=h, \quad u(a)=0$$

$$m v(1+\cos 2 \theta)$$

$$F=2 \pi n m v^2 \int_0^a r(1+\cos 2 \theta) d r$$

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