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Let $f \in C(S^{n},S^{n})$. If $n=1$ then the degree of $f$ coincides with index of curve $f(S^1)$ with respect to zero (winding number) and may be computed via integral $$ \deg f = \frac{1}{2\pi i} \int\limits_{f(S^1)} \frac{dz}{z} $$ Is it possible to compute the degree of continuous mapping $f$ in the case $n>1$ via integral of some differential form?

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The short answer is: Take the $n$-form $\omega$ that generates $H^n(S^n)$. Then $\deg f$ is the ratio of $\int f^* \omega$ over $\int \omega$. – user27126 Jan 26 '13 at 11:00

You could find some useful information (try page 6) here and here.

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