Let 𝐅 be the vortex vector field 𝐅(x, y)=⟨(-y)/(x^2+y^2), (x)/(x^2+y^2)⟩In Section 16.3, we veri

step 1 In this even question we have to prove that F .DR over C is equal to 2 pi where C is any simple

Let 𝐅 be the vortex vector field 𝐅(x, y)=⟨(-y)/(x^2+y^2),...

Let $\mathbf{F}$ be the vortex vector field $$\mathbf{F}(x, y)=\left\langle\frac{-y}{x^{2}+y^{2}}, \frac{x}{x^{2}+y^{2}}\right\rangle$$ In Section $16.3,$ we verified that $\int_{\mathcal{C}_{R}} \mathbf{F} \cdot d \mathbf{r}=2 \pi,$ where $\mathcal{C}_{R}$ is the circle of radius $R$ centered at the origin. Prove that $\oint_{\mathcal{C}} \mathbf{F} \cdot d \mathbf{r}=2 \pi$ for any simple closed curve $\mathcal{C}$ whose interior contains the origin (Figure 26$) .$ Hint: Apply the general form of Green's Theorem to the domain between $\mathcal{C}$ and $\mathcal{C}_{R},$ where $R$ is so small that $\mathcal{C}_{R}$ is contained in $\mathcal{C}$ .

Let $\mathbf{F}$ be the vortex vector field
$$\mathbf{F}(x, y)=\left\langle\frac{-y}{x^{2}+y^{2}}, \frac{x}{x^{2}+y^{2}}\right\rangle$$
In Section $16.3,$ we verified that $\int_{\mathcal{C}_{R}} \mathbf{F} \cdot d \mathbf{r}=2 \pi,$ where $\mathcal{C}_{R}$ is the circle of radius $R$ centered at the origin. Prove that $\oint_{\mathcal{C}} \mathbf{F} \cdot d \mathbf{r}=2 \pi$
for any simple closed curve $\mathcal{C}$ whose interior contains the origin (Figure 26$) .$ Hint: Apply the general form of Green's Theorem to the domain between $\mathcal{C}$ and $\mathcal{C}_{R},$ where $R$ is so small that $\mathcal{C}_{R}$ is contained in $\mathcal{C}$ .

Key Concepts

Green's Theorem

Green's Theorem is a fundamental result in vector calculus that relates a line integral around a simple closed curve to a double integral over the region bounded by the curve. It serves as a bridge between the circulation of a vector field along the boundary and the behavior of the field within the enclosed area. Its general form is particularly useful when dealing with regions that have holes or singularities since it can be extended to account for multiple boundary components with appropriate orientations.

Handling Singularities in Vector Fields

When a vector field has a singularity (a point where the field is not defined or becomes unbounded), such as at the origin in this example, one must carefully choose integration domains that avoid the singular point. By excluding the singularity via a small, additional closed curve, and then applying Green's Theorem on the region between the curves, one can correctly account for the impact of the singularity on the field's circulation or other integral properties.

Orientation and Multiply-Connected Regions

In regions with multiple boundary components, such as the area between an outer curve and a small inner curve surrounding a singularity, the orientation of each boundary must be carefully considered. The outer boundary is typically oriented counterclockwise, while the inner boundary, which represents a hole in the region, is oriented clockwise. This proper handling of orientations is essential when applying Green's Theorem to ensure that contributions from each part of the boundary are combined correctly.

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