### 统计代写|蒙特卡洛方法代写Monte Carlo method代考|ME 777

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Terminology

Further discussion of the nature of thermal radiation requires careful definition of certain concepts and terms of art. Chief among these is the concept of the solid angle $\Omega$ (sr), whose thorough understanding is critical to the study of both radiation heat transfer and applied optics. Consider Figure $2.3$, which shows a differential surface element $d S$ located a distance $r$ from a second differential area $d A$. The line of length $r$ connecting $d A$ and $d S$ intersects the normal to $d A$ at an angle $\beta_{A}$, and it intersects the normal to $d S$ at an angle $\beta_{S}$. As a concession to clarity, the surface elements $d A$ and $d S$ are necessarily drawn as finite in size but, in fact, both are arbitrarily small compared to the finite distance $r$. The surface element $d S \cos \beta_{S}$, which is hinged to surface element $d S$ along their common lower edge, is tilted toward $d A$, so that the line $r$ is normal to $d S \cos \beta_{S}$. Because both $d S$ and $d S \cos \beta_{S}$ are vanishingly small, the distance between the points where $r$ intersects them is negligible compared to the finite length $r$. Then the differential solid angle $d \Omega_{S}$ subtended by $d S \cos \beta_{S}$ at $d A$ is defined
$$d \Omega_{S} \equiv \frac{d S \cos \beta_{S}}{r^{2}} .$$
Note that the solid angle in steradians (sr) is actually a dimensionless ratio of area over length squared, just as a one-dimensional angle in radians (r) is a dimensionless ratio of lengths.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Intensity of Radiation

The term spectral and its synonym monochromatic (“mono” = one, “chrome” = color) refer to radiation confined to a vanishingly small wavelength interval $d \lambda$ centered about a specified wavelength $\lambda$. Thus, the polarized ray in Figure $2.1$ represents spectral radiation. The spectral intensity $i_{\lambda}(\lambda, \vartheta, \varphi)$ of a plane source is the power per unit wavelength in the wavelength interval $d \lambda$ centered about wavelength $\lambda$, per unit projected area of the source, per unit solid angle, passing in direction $(\vartheta, \varphi)$. Note that the symbol $\lambda$ appears twice in the notation. This is not

redundant usage; the subscript $\lambda$ signals that the spectral intensity is a per-unit-wavelength quantity, and the $\lambda$ in the argument list signals that the value of the spectral intensity depends on wavelength. While it is traditional to call this quantity “intensity” in the radiation heat transfer community, it is frequently referred to as “radiance” in the applied optics, astronomy, and earth sciences literature.

Figure $2.5$ represents a beam of monochromatic light of spectral power $d^{3} P(\lambda, \vartheta, \varphi)$ (W) leaving the plane surface element $d A$ in direction $(\vartheta, \varphi)$ at an angle $\vartheta$ with respect to the surface normal and contained in a beam whose solid angle is $d \Omega_{s}$. Then the spectral intensity of this beam is
$$i_{\lambda}(\lambda, \vartheta, \varphi)=\frac{d^{3} P(\lambda, \vartheta, \varphi)}{d A \cos \vartheta d \Omega_{S} d \lambda}\left(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{sr} \cdot \mu \mathrm{m}\right) .$$
The superscript ” 3 ” on the differential operator in the numerator of Eq. (2.7) is required for notational consistency; in order for the intensity to be a finite quantity, the number of differential symbols $d$ must be the same in both the numerator and the denominator.

Another useful concept is the total intensity of a beam, which is obtained by integrating the spectral intensity over all wavelengths, i.e.,
$$i(\vartheta, \varphi)=\int_{\lambda=0}^{\infty} i_{\lambda}(\lambda, \vartheta, \varphi) d \lambda .$$
The word “total” is used exclusively in this context in the radiation heat transfer literature.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Directional Spectral Emissive Power

The directional spectral emissive power $e_{\lambda}(\lambda, \vartheta, \varphi)$ of a plane source is the power per unit wavelength in a specified wavelength interval $d \lambda$ about wavelength $\lambda$, per unit source surface area $d A$, emitted in direction $(\vartheta, \varphi)$ per unit solid angle into the space above the source. Then referring once again to Figure $2.5$, the differential directional spectral emissive power contained in the solid angle $d \Omega_{s}$ is
$$d E_{\lambda} \equiv \frac{d^{3} P(\lambda, \vartheta, \varphi)}{d A d \lambda}\left(\mathrm{W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}\right) .$$
Invoking Eq. (2.7) we have
$$d E_{\lambda}=i_{\lambda}(\lambda, \vartheta, \phi) \cos \vartheta d \Omega_{s}\left(\mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}\right)$$
Finally, the directional spectral emissive power is
$$e_{\lambda}(\lambda, \vartheta, \varphi) \equiv \frac{d E_{\lambda}}{d \Omega_{\mathrm{S}}}=i_{\lambda}(\lambda, \vartheta, \phi) \cos \vartheta\left(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{sr} \cdot \mu \mathrm{m}\right) .$$

dΩ小号≡d小号因⁡b小号r2.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Directional Spectral Emissive Power

d和λ≡d3磷(λ,ϑ,披)d一个dλ(在/米2⋅μ米).

d和λ=一世λ(λ,ϑ,φ)因⁡ϑdΩs( 在/米2⋅μ米)

## 有限元方法代写

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## MATLAB代写

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