# 微积分网课代修|函数代写Function theory代考|MATH824 Calculus in the Complex Domain

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考Calculus in the Complex Domain

In order that we may be able to do calculus computations easily and efficiently in the context of complex analysis, we recast some of the basic ideas in new notation. We define the differential operators
$$\frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x}-i \frac{\partial}{\partial y}\right) \quad \text { and } \quad \frac{\partial}{\partial \bar{z}}=\frac{1}{2}\left(\frac{\partial}{\partial x}+i \frac{\partial}{\partial y}\right)$$
This is, in effect, a new basis for the tangent space to $\mathbb{C}$. In complex analysis it is more convenient to use these operators than to use $\partial / \partial x$ and $\partial / \partial y$.

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考|Isometries

In any mathematical subject there are morphisms: functions that preserve the relevant properties being studied. In linear algebra these are linear maps, in Euclidean geometry these are rigid motions, and in Riemannian geometry these are “isometries.” We now define the concept of isometry.
Definition 2.3.7. Let $\Omega_{1}$ and $\Omega_{2}$ be planar domains and let
$$f: \Omega_{1} \rightarrow \Omega_{2}$$
be a continuously differentiable mapping with Jacobian having isolated zeros. Assume that $\Omega_{2}$ is equipped with a metric $\rho$. We define the pullback of the metric $\rho$ under the map $f$ to be the metric on $\Omega_{1}$ given by
$$f^{*} \rho(z)=\rho(f(z)) \cdot\left|\frac{\partial f}{\partial z}\right| .$$

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考Calculus in the Complex Domain

$\frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x}-i \frac{\partial}{\partial y}\right) \quad$ and $\quad \frac{\partial}{\partial \bar{z}}=\frac{1}{2}\left(\frac{\partial}{\partial x}+i \frac{\partial}{\partial y}\right)$

## 微积分网课代修|偏微分方程代写Partial Differential Equation代 考|Isometries

$$f: \Omega_{1} \rightarrow \Omega_{2}$$

$$f^{*} \rho(z)=\rho(f(z)) \cdot\left|\frac{\partial f}{\partial z}\right| .$$