# 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|MATH745 Polar Coordinates

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## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|Polar Coordinates

There are several special “coordinate systems” that are important in the analysis of differential equations, especially, polar coordinates, cylindrical coordinates, and spherical coordinates. In this section we will consider the meaning of these coordinates in the language of differentiable manifolds, and we will also explore a few applications, especially blowup of a rest point and compactification at infinity. However, the main purpose of this section is to provide a deeper understanding and appreciation for the manifold concept in the context of the study of differential equations.
What are polar coordinates?
Perhaps the best way to understand the meaning of polar coordinates is to recall the “angular wrapping function” definition of angular measure from elementary trigonometry. We have proved that the unit circle $\mathbb{T}$ is a one-dimensional manifold. The wrapping function $P: \mathbb{R} \rightarrow \mathbb{T}$ is given by
$$P(\theta)=(\cos \theta, \sin \theta) .$$

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|Compactification at Infinity

The orbits of a differential equation on $\mathbb{R}^{n}$ may be unbounded. One way to obtain some information about the behavior of such solutions is to (try to) compactify the Cartesian space, so that the vector field is extended to a new manifold that contains the “points at infinity.” This idea, due to Henri Poincaré [143], has been most successful in the study of planar systems given by polynomial vector fields, also called polynomial systems (see $[5$, p. 219] and [76]). In this section we will give a brief description of the compactification process for such planar systems. We will again use the manifold concept and the idea of reparametrization.
Let us consider a plane vector field, which we will write in the form
$$\dot{x}=f(x, y), \quad \dot{y}=g(x, y) .$$

## 微积分网课代修常微分方程代写Ordinary Differential Equation代 考|Polar Coordinates

$$P(\theta)=(\cos \theta, \sin \theta) .$$

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代 考 Compactification at Infinity

$$\dot{x}=f(x, y), \quad \dot{y}=g(x, y)$$