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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 物理代写|广义相对论代写General relativity代考|TIIE COVARIANT DERIVATIVE OF VECTORS

Let us consider a vector (field) $\vec{V}=V^{\alpha} \vec{e}{(\alpha)}$ on a manifold $\mathbf{M}$, which we assume to be the spacetime, and let $\left{x^{\alpha}\right}$ be a chosen coordinate system. We want to compute the derivative of $\vec{V}$ with respect to the coordinates. By applying Leibniz’s rule we get $$\frac{\partial \vec{V}}{\partial x^{\beta}}=\frac{\partial V^{\alpha}}{\partial x^{\beta}} \vec{e}{(\alpha)}+V^{\alpha} \frac{\partial \vec{e}{(\alpha)}}{\partial x^{\beta}} .$$ The first term on the right-hand side is a linear combination of the basis vectors. The second term involves the derivative of the basis vectors, for which we need to compute the quantities $\vec{e}{(\alpha)}\left(\mathbf{p}^{\prime}\right)-\vec{e}{(\alpha)}(\mathbf{p})$, i.e. to subtract vectors which are applied at different points of the manifold. Note that the vectors $\vec{e}{(\alpha)}(\mathbf{p})$ and $\vec{e}{(\alpha)}\left(\mathbf{p}^{\prime}\right)$ belong, respectively, to $\mathbf{T}{\mathbf{p}}$ and to $\mathbf{T}{\mathbf{p}^{\prime}}$, and that $\mathbf{T}{\mathbf{p}} \neq \mathbf{T}_{\mathbf{p}^{\prime}}$, since $\mathbf{p}$ and $\mathbf{p}^{\prime}$ are distinct. Thus, to define the derivative of a vector field on a manifold, we need to specify a rule to compare vectors belonging to different tangent spaces; this rule is called connection.

Let us start by considering Minkowski’s spacetime, where it is possible to define a global Minkowskian coordinate system $\left{\xi^{\mu}\right}=(c t, x, y, z)$ which covers the entire spacetime; at any given point $\mathbf{p}$ of the manifold there exists the coordinate basis $\vec{e}{M(\alpha)}(\mathbf{p})$ which belongs to the tangent space $\mathbf{T}{\mathbf{p}}$, which is the same for any $\mathbf{p}$. In this case a simple rule to compare vectors at different points is to impose that each basis vector at a point $\mathbf{p}$ is equal to the corresponding basis vector at any other point $\mathbf{p}^{\prime}$, i.e.
$$\vec{e}{M(\alpha)}(\mathbf{p})=\vec{e}{M(\alpha)}\left(\mathbf{p}^{\prime}\right) .$$

## 物理代写|广义相对论代写General relativity代考|THE COVARIANT DERIVATIVE OF SCALARS AND ONE-FORMS

Let us consider a scalar field $\Phi$. At any given point $\mathbf{p}$ of the manifold, $\Phi(\mathbf{p})$ is a real number, the value of which does not depend on the choice of the coordinate system. However, $\Phi(\mathbf{p})$ has a specific dependence on the chosen coordinates. Therefore $\Phi(\mathbf{p})=\Phi\left(x^{\mu}\right)$ is a real function of the coordinates.

Since a scalar function does not depend on the basis vectors, the covariant derivative of a scalar field on a manifold coincides with the ordinary derivative:
$$\nabla_{\mu} \Phi \equiv \frac{\partial \Phi}{\partial x^{\mu}} .$$

We remind that, as shown at the end of Sec. 2.3, the differential of a function $\Phi$ is a one-form whose components are
$$d \Phi_{\mu}=\frac{\partial \Phi}{\partial x^{\mu}},$$
and that we have adopted the convention to omit the tilde over the differential $d \Phi$. Thus, the
In order to define the covariant derivative of a one-form field $\tilde{q}=q_{\alpha} \tilde{\omega}^{(\alpha)}$, we may proceed as in Sec. $3.1$ assuming that, due to the Equivalence Principle, the first derivatives of the basis one-forms in a LIF vanish,
$$\frac{\partial \tilde{\omega}_{M}^{\left(\mu^{\prime}\right)}}{\partial \xi^{\beta^{\prime}}}=\tilde{0} .$$
However, we shall follow a simpler derivation, based on Eq. $3.28$ and on the fact that derivative operators have to satisfy Leibniz’s rule.

The one-form field $\tilde{q}$ is, by definition, a linear, real valued function of vectors such that
$$\tilde{q}(\vec{V})=q_{\alpha} V^{\alpha} .$$

# 广义相对论代考

## 物理代写|广义相对论代写General relativity代考|TIIE COVARIANT DERIVATIVE OF VECTORS

$$\frac{\partial \vec{V}}{\partial x^{\beta}}=\frac{\partial V^{\alpha}}{\partial x^{\beta}} \vec{e}(\alpha)+V^{\alpha} \frac{\partial \vec{e}(\alpha)}{\partial x^{\beta}} .$$

$$\vec{e} M(\alpha)(\mathbf{p})=\vec{e} M(\alpha)\left(\mathbf{p}^{\prime}\right) .$$

## 物理代写|广义相对论代写General relativity代考|THE COVARIANT DERIVATIVE OF SCALARS AND ONE-FORMS

$$\nabla_{\mu} \Phi \equiv \frac{\partial \Phi}{\partial x^{\mu}} .$$

$$d \Phi_{\mu}=\frac{\partial \Phi}{\partial x^{\mu}},$$

$$\frac{\partial \tilde{\omega}{M}^{\left(\mu^{\prime}\right)}}{\partial \xi^{\beta^{\prime}}}=\tilde{0}$$ 但是，我们将遵循基于方程式的更简单的推导。3.28以及导数运算符必须满足莱布尼茨规则的事实。 单式字段 $\tilde{q}$ 根据定义，是向量的线性实值函数，使得 $$\tilde{q}(\vec{V})=q{\alpha} V^{\alpha} .$$

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师