The winding number and index of curve

If $\gamma$ is a smooth closed curve in $\mathbb R^2-\{0\}$, I want to know whether the winding number of $\gamma$ about $0$, i.e., $\frac{1}{{2\pi i}}\int\limits_\gamma {\frac{1}{z}} dz$ is equal to the index of $\gamma$ about $0$, that is, the degree of ${S^1} \to {S^1}$ through $t \to \frac{{\gamma (t)}}{{\left| {\gamma (t)} \right|}}$?

If so, it gives us a convenient way to calculate the index of curve. Is there any analogy in higher dimension?

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Yes, the index of a curve around $0$ is equal to the winding number $\displaystyle\frac{1}{2\pi i}\oint_\gamma \frac{dz}{z}$. Note that this can also be written as a line integral: $\displaystyle\frac{1}{2\pi}\oint_\gamma \frac{-y\,dx+x\,dy}{r^2}$, where $r^2=x^2+y^2$.
There are two simple generalizations of this formula to three dimensions. First, if $S$ is any closed paramterized surface, then we can compute the index of $S$ around the origin using the surface integral $$\frac{1}{4\pi}\oint_S \frac{\textbf{r}\cdot\textbf{n}}{r^3}\;dA,$$ where $\textbf{n}$ is the normal vector to the surface, $\textbf{r}$ is the vector $(x,y,z)$, and $r=\sqrt{x^2+y^2+z^2}$. This formula generalizes to $n$ dimensions, with $4\pi$ replaced by the $(n-1)$-dimensional area of the unit sphere in $\mathbb{R}^n$, and $r^3$ denominator replaced by $r^n$.
The other generalization is the integral formula for the linking number of two closed curves. If $\gamma_1$ and $\gamma_2$ are disjoint closed curves in $\mathbb{R}^3$, their linking number is given by the following double line integral: $$\frac{1}{4\pi}\oint_{\gamma_1}\oint_{\gamma_2} \frac{\textbf{r}_1-\textbf{r}_2}{\|\textbf{r}_1-\textbf{r}_2\|^3}\cdot(d\textbf{r}_1\times d\textbf{r}_2)$$ In the case where one of the curves is the $z$-axis, this presumably simplifies to the line integral for winding number given above.