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Introduction

I am a mathematics amateur and I am learning the book "Visual Differential Geometry and Forms" in my leisure time.

Since I am not professional, I am publishing my solutions here and I am willing to accept your criticism. If you find there are any mistakes or improvements, please don't hesitate to contact me by email.

Prologue: Newtonian Ultimate Equality ( ≍ )

3-1

The ultimate equality notation "≍" actually represents the equivalent infinitesimal which can be checked by the Taylor's theorem.

Act I: Euclidean and Non-Euclidean Geometry

The most important point of the example of the negative curvature is to wave up and down simultaneously.

3-6

By (1.3), we have $\displaystyle \lim\limits_{\vartriangle_p \rightarrow 0} \operatorname{\varepsilon}(\vartriangle_p) = \lim\limits_{\vartriangle_p \rightarrow 0}\frac{\operatorname{A}(\vartriangle_p)}{R^2} = \frac{1}{R^2}\lim\limits_{\vartriangle_p \rightarrow 0}\operatorname{A}(\vartriangle_p) = \frac{1}{R^2} \times 0 = 0$.

And since "ε ≡ (angle sum of triangle) − π", we have $\displaystyle \lim\limits_{\vartriangle_p \rightarrow 0} \text{(angle sum of triangle)} − \pi = 0$ and thus $\displaystyle \lim\limits_{\vartriangle_p \rightarrow 0} \text{(angle sum of triangle)} = \pi$.