list-of-mathstuff.qmd

\providecommand{\Pageref}[1]{\hfill p.\pageref{#1}}
 [**Goal**](#Goal*-1.1)\Pageref{Goal*-1.1}

 [**Question**](#Question*-1.2)\Pageref{Question*-1.2}

 [**Problem**](#Problem*-1.3)\Pageref{Problem*-1.3}

 [**Definition 1.1**](#parametrized-curve): Parametrized curve\Pageref{parametrized-curve}

 [**Definition 1.2**](#level-curve): Level curve\Pageref{level-curve}

 [**Definition 1.3**](#Definition-1.3)\Pageref{Definition-1.3}

 [**Question**](#Question*-1.4)\Pageref{Question*-1.4}

 [**Example 1.4**](#parametrizing-the-straight-line): Parametrizing the straight line\Pageref{parametrizing-the-straight-line}

 [**Remark 1.5**](#parametrization-is-not-unique): Parametrization is not unique\Pageref{parametrization-is-not-unique}

 [**Example 1.6**](#parametrizing-the-circle): Parametrizing the circle\Pageref{parametrizing-the-circle}

 [**Example 1.7**](#parametrizing-the-helix): Parametrizing the helix\Pageref{parametrizing-the-helix}

 [**Definition 1.8**](#parametrized-curve-1): Parametrized curve\Pageref{parametrized-curve-1}

 [**Definition 1.9**](#smooth-functions): Smooth functions\Pageref{smooth-functions}

 [**Example 1.10**](#Example-1.10)\Pageref{Example-1.10}

 [**Definition 1.11**](#Definition-1.11)\Pageref{Definition-1.11}

 [**Assumption**](#Assumption*-1.5)\Pageref{Assumption*-1.5}

 [**Example 1.12**](#Example-1.12)\Pageref{Example-1.12}

 [**Definition 1.13**](#definition-tangent): Tangent vector\Pageref{definition-tangent}

 [**Example 1.14**](#tangent-vector-to-helix): Tangent vector to helix\Pageref{tangent-vector-to-helix}

 [**Remark 1.15**](#avoiding-potential-ambiguities): Avoiding potential ambiguities\Pageref{avoiding-potential-ambiguities}

 [**Example 1.16**](#the-lemniscate-a-self-intersecting-curve): The Lemniscate, a self intersecting curve\Pageref{the-lemniscate-a-self-intersecting-curve}

 [**Definition 1.17**](#partition): Partition\Pageref{partition}

 [**Definition 1.18**](#length-of-approximating-polygonal-curve): Length of approximating polygonal curve\Pageref{length-of-approximating-polygonal-curve}

 [**Definition 1.19**](#rectifiable-curve-and-length): Rectifiable curve and length\Pageref{rectifiable-curve-and-length}

 [**Question 1.20**](#Question-1.20)\Pageref{Question-1.20}

 [**Theorem 1.21**](#theorem-length): Characterizing the length of ${\pmb{\gamma}}$\Pageref{theorem-length}

 [**Proof**](#Proof*-1.6)\Pageref{Proof*-1.6}

 [**Example 1.22**](#length-of-circle): Length of Circle\Pageref{length-of-circle}

 [**Example 1.23**](#length-of-helix): Length of helix\Pageref{length-of-helix}

 [**Definition 1.24**](#arc-length-1): Arc-length\Pageref{arc-length-1}

 [**Remark 1.25**](#Remark-1.25)\Pageref{Remark-1.25}

 [**Example 1.26**](#circle): Circle\Pageref{circle}

 [**Example 1.27**](#logarithmic-spiral): Logarithmic spiral\Pageref{logarithmic-spiral}

 [**Definition 1.28**](#scalar-product-in-mathbbr2): Scalar product in $\mathbb{R}^2$\Pageref{scalar-product-in-mathbbr2}

 [**Remark 1.29**](#Remark-1.29)\Pageref{Remark-1.29}

 [**Definition 1.30**](#orthogonal-vectors): Orthogonal vectors\Pageref{orthogonal-vectors}

 [**Proposition 1.31**](#bilinearity-and-symmetry-of-scalar-product): Bilinearity and symmetry of scalar product\Pageref{bilinearity-and-symmetry-of-scalar-product}

 [**Proposition 1.32**](#scalar-products-written-wrt-euclidean-coordinates): Scalar products written wrt euclidean coordinates\Pageref{scalar-products-written-wrt-euclidean-coordinates}

 [**Proof**](#Proof*-1.7)\Pageref{Proof*-1.7}

 [**Definition 1.33**](#scalar-product-in-mathbbrn-1): Scalar product in $\mathbb{R}^n$\Pageref{scalar-product-in-mathbbrn-1}

 [**Proposition 1.34**](#bilinearity-and-symmetry-of-scalar-product-in-mathbbrn): Bilinearity and symmetry of scalar product in $\mathbb{R}^n$\Pageref{bilinearity-and-symmetry-of-scalar-product-in-mathbbrn}

 [**Definition 1.35**](#Definition-1.35)\Pageref{Definition-1.35}

 [**Proposition 1.36**](#proposition-diff-scalar): Differentiating scalar product\Pageref{proposition-diff-scalar}

 [**Proof**](#Proof*-1.8)\Pageref{Proof*-1.8}

 [**Definition 1.37**](#Definition-1.37)\Pageref{Definition-1.37}

 [**Remark 1.38**](#Remark-1.38)\Pageref{Remark-1.38}

 [**Proposition 1.39**](#proposition-speed-identity)\Pageref{proposition-speed-identity}

 [**Proof**](#Proof*-1.9)\Pageref{Proof*-1.9}

 [**Remark 1.40**](#Remark-1.40)\Pageref{Remark-1.40}

 [**Definition 1.41**](#diffeomorphism): Diffeomorphism\Pageref{diffeomorphism}

 [**Definition 1.42**](#reparametrization-1): Reparametrization\Pageref{reparametrization-1}

 [**Remark 1.43**](#Remark-1.43)\Pageref{Remark-1.43}

 [**Example 1.44**](#change-of-orientation): Change of orientation\Pageref{change-of-orientation}

 [**Example 1.45**](#reversing-orientation-of-circle): Reversing orientation of circle\Pageref{reversing-orientation-of-circle}

 [**Example 1.46**](#change-of-speed): Change of speed\Pageref{change-of-speed}

 [**Example 1.47**](#doubling-the-speed-of-lemniscate): Doubling the speed of Lemniscate\Pageref{doubling-the-speed-of-lemniscate}

 [**Important**](#Important*-1.10)\Pageref{Important*-1.10}

 [**Definition 1.48**](#regular-points): Regular points\Pageref{regular-points}

 [**Lemma 1.49**](#lemma-diffeomorphism)\Pageref{lemma-diffeomorphism}

 [**Proof**](#Proof*-1.11)\Pageref{Proof*-1.11}

 [**Example 1.50**](#example-stopping-curve): A curve with one singular point\Pageref{example-stopping-curve}

 [**Proposition 1.51**](#proposition-regularity-invariant): Regularity is invariant for reparametrization\Pageref{proposition-regularity-invariant}

 [**Proof**](#Proof*-1.12)\Pageref{Proof*-1.12}

 [**Example 1.52**](#Example-1.52)\Pageref{Example-1.52}

 [**Definition 1.53**](#unit-speed-reparametrization): Unit speed reparametrization\Pageref{unit-speed-reparametrization}

 [**Theorem 1.54**](#theorem-regular-unit-speed): Existence of unit speed reparametrization\Pageref{theorem-regular-unit-speed}

 [**Proof**](#Proof*-1.13)\Pageref{Proof*-1.13}

 [**Proposition 1.55**](#proposition-arc-length-unit): Arc-length and unit speed reparametrization\Pageref{proposition-arc-length-unit}

 [**Proof**](#Proof*-1.14)\Pageref{Proof*-1.14}

 [**Remark 1.56**](#Remark-1.56)\Pageref{Remark-1.56}

 [**Example 1.57**](#twisted-cubic): Twisted cubic\Pageref{twisted-cubic}

 [**Example 1.58**](#example-curve-loop)\Pageref{example-curve-loop}

 [**Definition 1.59**](#periodic-curve): Periodic curve\Pageref{periodic-curve}

 [**Definition 1.60**](#closed-curve): Closed curve\Pageref{closed-curve}

 [**Remark 1.61**](#remark-closed)\Pageref{remark-closed}

 [**Definition 1.62**](#Definition-1.62)\Pageref{Definition-1.62}

 [**Proposition 1.63**](#Proposition-1.63)\Pageref{Proposition-1.63}

 [**Proof**](#Proof*-1.15)\Pageref{Proof*-1.15}

 [**Example 1.64**](#Example-1.64)\Pageref{Example-1.64}

 [**Definition 1.65**](#self-intersecting-curve): Self-intersecting curve\Pageref{self-intersecting-curve}

 [**Remark 1.66**](#Remark-1.66)\Pageref{Remark-1.66}

 [**Example 1.67**](#Example-1.67)\Pageref{Example-1.67}

 [**Example 1.68**](#the-limaçon): The Limaçon\Pageref{the-limaçon}

 [**Important**](#Important*-2.1)\Pageref{Important*-2.1}

 [**Definition 2.1**](#Definition-2.1)\Pageref{Definition-2.1}

 [**Definition 2.2**](#Definition-2.2)\Pageref{Definition-2.2}

 [**Remark 2.3**](#Remark-2.3)\Pageref{Remark-2.3}

 [**Remark 2.4**](#methods-for-computing-curvature): Methods for computing curvature\Pageref{methods-for-computing-curvature}

 [**Example 2.5**](#Example-2.5)\Pageref{Example-2.5}

 [**Remark 2.6**](#remark-hyperbolic-functions): Hyperbolic functions\Pageref{remark-hyperbolic-functions}

 [**Example 2.7**](#the-catenary): The Catenary\Pageref{the-catenary}

 [**Definition 2.8**](#same-orientation): Same orientation\Pageref{same-orientation}

 [**Definition 2.9**](#orientation): Orientation\Pageref{orientation}

 [**Definition 2.10**](#positive-orientation): Positive orientation\Pageref{positive-orientation}

 [**Example 2.11**](#Example-2.11)\Pageref{Example-2.11}

 [**Definition 2.12**](#vector-product-in-mathbbr3-1): Vector product in $\mathbb{R}^3$\Pageref{vector-product-in-mathbbr3-1}

 [**Proposition 2.13**](#Proposition-2.13)\Pageref{Proposition-2.13}

 [**Proof**](#Proof*-2.2)\Pageref{Proof*-2.2}

 [**Proposition 2.14**](#proposition-vector-product)\Pageref{proposition-vector-product}

 [**Remark 2.15**](#remark-vector-product): Geometric interpretation of vector product\Pageref{remark-vector-product}

 [**Remark 2.16**](#summary-properties-of-mathbfutimes-mathbfv): Summary: Properties of $\mathbf{u}\times \mathbf{v}$\Pageref{summary-properties-of-mathbfutimes-mathbfv}

 [**Proposition 2.17**](#Proposition-2.17)\Pageref{Proposition-2.17}

 [**Proposition 2.18**](#Proposition-2.18)\Pageref{Proposition-2.18}

 [**Proposition 2.19**](#proposition-curvature-formula): Curvature formula\Pageref{proposition-curvature-formula}

 [**Example 2.20**](#Example-2.20)\Pageref{Example-2.20}

 [**Example 2.21**](#Example-2.21)\Pageref{Example-2.21}

 [**Definition 2.22**](#Definition-2.22)\Pageref{Definition-2.22}

 [**Definition 2.23**](#Definition-2.23)\Pageref{Definition-2.23}

 [**Remark 2.24**](#Remark-2.24)\Pageref{Remark-2.24}

 [**Remark 2.25**](#Remark-2.25)\Pageref{Remark-2.25}

 [**Definition 2.26**](#Definition-2.26)\Pageref{Definition-2.26}

 [**Theorem 2.27**](#theorem-characterization-plane): Characterization of plane curves\Pageref{theorem-characterization-plane}

 [**Example 2.28**](#Example-2.28)\Pageref{Example-2.28}

 [**Definition 2.29**](#principal-normal-vector): Principal normal vector\Pageref{principal-normal-vector}

 [**Remark 2.30**](#Remark-2.30)\Pageref{Remark-2.30}

 [**Question 2.31**](#Question-2.31)\Pageref{Question-2.31}

 [**Definition 2.32**](#binormal-vector): Binormal vector\Pageref{binormal-vector}

 [**Definition 2.33**](#orthonormal-basis): Orthonormal basis\Pageref{orthonormal-basis}

 [**Proposition 2.34**](#Proposition-2.34)\Pageref{Proposition-2.34}

 [**Proof**](#Proof*-2.3)\Pageref{Proof*-2.3}

 [**Proposition 2.35**](#proposition-positive-basis)\Pageref{proposition-positive-basis}

 [**Proof**](#Proof*-2.4)\Pageref{Proof*-2.4}

 [**Proposition 2.36**](#proposition-binormal)\Pageref{proposition-binormal}

 [**Proof**](#Proof*-2.5)\Pageref{Proof*-2.5}

 [**Definition 2.37**](#torsion-of-unit-speed-curve): Torsion of unit speed curve\Pageref{torsion-of-unit-speed-curve}

 [**Remark 2.38**](#Remark-2.38)\Pageref{Remark-2.38}

 [**Warning**](#Warning*-2.6)\Pageref{Warning*-2.6}

 [**Definition 2.39**](#Definition-2.39)\Pageref{Definition-2.39}

 [**Proposition 2.40**](#proposition-torsion-formula): Torsion formula\Pageref{proposition-torsion-formula}

 [**Example 2.41**](#torsion-helix): Torsion Helix\Pageref{torsion-helix}

 [**Example 2.42**](#example-circle-curvature-torsion): Curvature and Torsion of Circle\Pageref{example-circle-curvature-torsion}

 [**Example 2.43**](#example-circle-rotated)\Pageref{example-circle-rotated}

 [**Definition 2.44**](#frenet-frame-1): Frenet frame\Pageref{frenet-frame-1}

 [**Definition 2.45**](#Definition-2.45)\Pageref{Definition-2.45}

 [**Remark 2.46**](#Remark-2.46)\Pageref{Remark-2.46}

 [**Theorem 2.47**](#frenet-serret-equations): Frenet-Serret equations\Pageref{frenet-serret-equations}

 [**Proof**](#Proof*-2.7)\Pageref{Proof*-2.7}

 [**Remark 2.48**](#Remark-2.48)\Pageref{Remark-2.48}

 [**Important**](#summary): Summary\Pageref{summary}

 [**Example 2.49**](#frenet-frame-of-helix): Frenet frame of helix\Pageref{frenet-frame-of-helix}

 [**Theorem 2.50**](#theorem-characterization-space): Characterization of space curves\Pageref{theorem-characterization-space}

 [**Example 2.51**](#Example-2.51)\Pageref{Example-2.51}

 [**Proposition 2.52**](#proposition-curvature-torsion-formulas): Curvature and torsion formulas\Pageref{proposition-curvature-torsion-formulas}

 [**Notation**](#compact-notation-for-arc-length-reparametrization): Compact notation for arc length reparametrization\Pageref{compact-notation-for-arc-length-reparametrization}

 [**Example 2.53**](#how-to-use-the-new-notations): How to use the new notations\Pageref{how-to-use-the-new-notations}

 [**Important**](#new-notation): New Notation!\Pageref{new-notation}

 [**Proof**](#proof-of-proposition): Proof of Proposition 2.52\Pageref{proof-of-proposition}

 [**Remark 2.54**](#equation-of-a-plane): Equation of a plane\Pageref{equation-of-a-plane}

 [**Proposition 2.55**](#proposition-zero-torsion)\Pageref{proposition-zero-torsion}

 [**Proof**](#Proof*-2.12)\Pageref{Proof*-2.12}

 [**Proposition 2.56**](#proposition-constant-k-zero-t)\Pageref{proposition-constant-k-zero-t}

 [**Proof**](#Proof*-2.13)\Pageref{Proof*-2.13}

 [**Definition 3.1**](#definition-topology): Topological space\Pageref{definition-topology}

 [**Remark 3.2**](#remark-intersection)\Pageref{remark-intersection}

 [**Warning**](#Warning*-3.1)\Pageref{Warning*-3.1}

 [**Definition 3.3**](#trivial-topology): Trivial topology\Pageref{trivial-topology}

 [**Definition 3.4**](#discrete-topology): Discrete topology\Pageref{discrete-topology}

 [**Definition 3.5**](#open-set-of-mathbbrn): Open set of $\mathbb{R}^n$\Pageref{open-set-of-mathbbrn}

 [**Definition 3.6**](#definition-euclidean-topology): Euclidean topology of $\mathbb{R}^n$\Pageref{definition-euclidean-topology}

 [**Proof**](#well-posedness-of-definition): Well-posedness of Definition 3.6\Pageref{well-posedness-of-definition}

 [**Proposition 3.7**](#proposition-balls-open)\Pageref{proposition-balls-open}

 [**Proof**](#Proof*-3.3)\Pageref{Proof*-3.3}

 [**Definition 3.8**](#closed-set): Closed set\Pageref{closed-set}

 [**Warning**](#Warning*-3.4)\Pageref{Warning*-3.4}

 [**Proposition 3.9**](#Proposition-3.9)\Pageref{Proposition-3.9}

 [**Proof**](#Proof*-3.5)\Pageref{Proof*-3.5}

 [**Example 3.10**](#the-zariski-topology): The Zariski topology\Pageref{the-zariski-topology}

 [**Definition 3.11**](#finer-and-coarser-topology): Finer and coarser topology\Pageref{finer-and-coarser-topology}

 [**Example 3.12**](#Example-3.12)\Pageref{Example-3.12}

 [**Example 3.13**](#cofinite-topology-on-mathbbr): Cofinite topology on $\mathbb{R}$\Pageref{cofinite-topology-on-mathbbr}

 [**Definition 3.14**](#convergent-sequence): Convergent sequence\Pageref{convergent-sequence}

 [**Notation**](#Notation*-3.6)\Pageref{Notation*-3.6}

 [**Proposition 3.15**](#convergence-for-trivial-topology): Convergence for trivial topology\Pageref{convergence-for-trivial-topology}

 [**Proof**](#Proof*-3.7)\Pageref{Proof*-3.7}

 [**Warning**](#Warning*-3.8)\Pageref{Warning*-3.8}

 [**Proposition 3.16**](#convergence-for-discrete-topology): Convergence for discrete topology\Pageref{convergence-for-discrete-topology}

 [**Proof**](#Proof*-3.9)\Pageref{Proof*-3.9}

 [**Definition 3.17**](#classical-convergence-in-mathbbrn): Classical convergence in $\mathbb{R}^n$\Pageref{classical-convergence-in-mathbbrn}

 [**Proposition 3.18**](#proposition-convergence-Rn): Convergence for Euclidean topology\Pageref{proposition-convergence-Rn}

 [**Proof**](#Proof*-3.10)\Pageref{Proof*-3.10}

 [**Notation**](#Notation*-3.11)\Pageref{Notation*-3.11}

 [**Proposition 3.19**](#Proposition-3.19)\Pageref{Proposition-3.19}

 [**Proof**](#Proof*-3.12)\Pageref{Proof*-3.12}

 [**Definition 3.20**](#distance): Distance\Pageref{distance}

 [**Definition 3.21**](#metric-space): Metric space\Pageref{metric-space}

 [**Example 3.22**](#mathbbrn-as-metric-space): $\mathbb{R}^n$ as metric space\Pageref{mathbbrn-as-metric-space}

 [**Example 3.23**](#p-distance-on-mathbbrn): $p$-distance on $\mathbb{R}^n$\Pageref{p-distance-on-mathbbrn}

 [**Definition 3.24**](#definition-metric-topology): Topology induced by the metric\Pageref{definition-metric-topology}

 [**Example 3.25**](#topology-induced-by-euclidean-distance): Topology induced by Euclidean distance\Pageref{topology-induced-by-euclidean-distance}

 [**Example 3.26**](#discrete-distance): Discrete distance\Pageref{discrete-distance}

 [**Proposition 3.27**](#proposition-metric-ball)\Pageref{proposition-metric-ball}

 [**Definition 3.28**](#equivalent-metrics): Equivalent metrics\Pageref{equivalent-metrics}

 [**Proposition 3.29**](#proposition-equivalent-metrics)\Pageref{proposition-equivalent-metrics}

 [**Example 3.30**](#Example-3.30)\Pageref{Example-3.30}

 [**Warning**](#Warning*-3.13)\Pageref{Warning*-3.13}

 [**Proposition 3.31**](#convergence-in-metric-space): Convergence in metric space\Pageref{convergence-in-metric-space}

 [**Definition 3.32**](#interior-of-a-set): Interior of a set\Pageref{interior-of-a-set}

 [**Remark 3.33**](#Remark-3.33)\Pageref{Remark-3.33}

 [**Proposition 3.34**](#Proposition-3.34)\Pageref{Proposition-3.34}

 [**Proof**](#Proof*-3.14)\Pageref{Proof*-3.14}

 [**Definition 3.35**](#closure-of-a-set): Closure of a set\Pageref{closure-of-a-set}

 [**Remark 3.36**](#Remark-3.36)\Pageref{Remark-3.36}

 [**Proposition 3.37**](#Proposition-3.37)\Pageref{Proposition-3.37}

 [**Proof**](#Proof*-3.15)\Pageref{Proof*-3.15}

 [**Lemma 3.38**](#lemma-closure-intersection)\Pageref{lemma-closure-intersection}

 [**Proof**](#Proof*-3.16)\Pageref{Proof*-3.16}

 [**Definition 3.39**](#boundary-of-a-set): Boundary of a set\Pageref{boundary-of-a-set}

 [**Proposition 3.40**](#Proposition-3.40)\Pageref{Proposition-3.40}

 [**Proof**](#Proof*-3.17)\Pageref{Proof*-3.17}

 [**Definition 3.41**](#Definition-3.41)\Pageref{Definition-3.41}

 [**Proposition 3.42**](#proposition-closure-sequence)\Pageref{proposition-closure-sequence}

 [**Proof**](#Proof*-3.18)\Pageref{Proof*-3.18}

 [**Warning**](#Warning*-3.19)\Pageref{Warning*-3.19}

 [**Example 3.43**](#example-closure-counter): Co-countable topology\Pageref{example-closure-counter}

 [**Proposition 3.44**](#proposition-closure-metric)\Pageref{proposition-closure-metric}

 [**Proof**](#Proof*-3.20)\Pageref{Proof*-3.20}

 [**Example 3.45**](#Example-3.45)\Pageref{Example-3.45}

 [**Definition 3.46**](#density-1): Density\Pageref{density-1}

 [**Proposition 3.47**](#Proposition-3.47)\Pageref{Proposition-3.47}

 [**Proof**](#Proof*-3.21)\Pageref{Proof*-3.21}

 [**Example 3.48**](#Example-3.48)\Pageref{Example-3.48}

 [**Example 3.49**](#Example-3.49)\Pageref{Example-3.49}

 [**Definition 3.50**](#Definition-3.50)\Pageref{Definition-3.50}

 [**Proposition 3.51**](#proposition-hausdorff-1)\Pageref{proposition-hausdorff-1}

 [**Proof**](#Proof*-3.22)\Pageref{Proof*-3.22}

 [**Definition 3.52**](#metrizable-space): Metrizable space\Pageref{metrizable-space}

 [**Corollary 3.53**](#corollary-hausdorff)\Pageref{corollary-hausdorff}

 [**Proof**](#Proof*-3.23)\Pageref{Proof*-3.23}

 [**Example 3.54**](#trivial-topology-is-not-hausdorff): Trivial topology is not Hausdorff\Pageref{trivial-topology-is-not-hausdorff}

 [**Example 3.55**](#cofinite-topology-on-mathbbr-1): Cofinite topology on $\mathbb{R}$\Pageref{cofinite-topology-on-mathbbr-1}

 [**Example 3.56**](#Example-3.56)\Pageref{Example-3.56}

 [**Proposition 3.57**](#uniqueness-of-limit-in-hausdorff-spaces): Uniqueness of limit in Hausdorff spaces\Pageref{uniqueness-of-limit-in-hausdorff-spaces}

 [**Proof**](#Proof*-3.24)\Pageref{Proof*-3.24}

 [**Definition 3.58**](#images-and-pre-images): Images and Pre-images\Pageref{images-and-pre-images}

 [**Warning**](#Warning*-3.25)\Pageref{Warning*-3.25}

 [**Proposition 3.59**](#Proposition-3.59)\Pageref{Proposition-3.59}

 [**Definition 3.60**](#continuous-function): Continuous function\Pageref{continuous-function}

 [**Proposition 3.61**](#proposition-continuity-characterization)\Pageref{proposition-continuity-characterization}

 [**Important**](#Important*-3.26)\Pageref{Important*-3.26}

 [**Example 3.62**](#Example-3.62)\Pageref{Example-3.62}

 [**Definition 3.63**](#continuity-in-the-classical-sense): Continuity in the classical sense\Pageref{continuity-in-the-classical-sense}

 [**Proposition 3.64**](#Proposition-3.64)\Pageref{Proposition-3.64}

 [**Proof**](#Proof*-3.27)\Pageref{Proof*-3.27}

 [**Proposition 3.65**](#Proposition-3.65)\Pageref{Proposition-3.65}

 [**Example 3.66**](#Example-3.66)\Pageref{Example-3.66}

 [**Example 3.67**](#Example-3.67)\Pageref{Example-3.67}

 [**Proposition 3.68**](#proposition-continuity-sequential)\Pageref{proposition-continuity-sequential}

 [**Proof**](#Proof*-3.28)\Pageref{Proof*-3.28}

 [**Warning**](#Warning*-3.29)\Pageref{Warning*-3.29}

 [**Proposition 3.69**](#proposition-sequential-continuity)\Pageref{proposition-sequential-continuity}

 [**Proof**](#Proof*-3.30)\Pageref{Proof*-3.30}

 [**Example 3.70**](#example-sequential-continuity)\Pageref{example-sequential-continuity}

 [**Proposition 3.71**](#Proposition-3.71)\Pageref{Proposition-3.71}

 [**Proof**](#Proof*-3.31)\Pageref{Proof*-3.31}

 [**Definition 3.72**](#homeomoprhim): Homeomoprhim\Pageref{homeomoprhim}

 [**Definition 3.73**](#definition-subspace-topology): Subspace topology\Pageref{definition-subspace-topology}

 [**Proof**](#well-posedness-of-definition-1): Well-posedness of Definition 3.73\Pageref{well-posedness-of-definition-1}

 [**Proposition 3.74**](#Proposition-3.74)\Pageref{Proposition-3.74}

 [**Proof**](#Proof*-3.33)\Pageref{Proof*-3.33}

 [**Warning**](#Warning*-3.34)\Pageref{Warning*-3.34}

 [**Example 3.75**](#Example-3.75)\Pageref{Example-3.75}

 [**Definition 3.76**](#topological-basis-1): Topological basis\Pageref{topological-basis-1}

 [**Example 3.77**](#Example-3.77)\Pageref{Example-3.77}

 [**Proposition 3.78**](#proposition-basis)\Pageref{proposition-basis}

 [**Proof**](#Proof*-3.35)\Pageref{Proof*-3.35}

 [**Proposition 3.79**](#proposition-topology-basis)\Pageref{proposition-topology-basis}

 [**Proof**](#Proof*-3.36)\Pageref{Proof*-3.36}

 [**Proposition 3.80**](#Proposition-3.80)\Pageref{Proposition-3.80}

 [**Definition 3.81**](#product-topology-1): Product topology\Pageref{product-topology-1}

 [**Example 3.82**](#Example-3.82)\Pageref{Example-3.82}

 [**Proposition 3.83**](#Proposition-3.83)\Pageref{Proposition-3.83}

 [**Proof**](#Proof*-3.37)\Pageref{Proof*-3.37}

 [**Proposition 3.84**](#Proposition-3.84)\Pageref{Proposition-3.84}

 [**Definition 3.85**](#connected-space): Connected space\Pageref{connected-space}

 [**Definition 3.86**](#proper-subset): Proper subset\Pageref{proper-subset}

 [**Proposition 3.87**](#proposition-connected-equivalent): Equivalent definition for connectedness\Pageref{proposition-connected-equivalent}

 [**Proof**](#Proof*-3.38)\Pageref{Proof*-3.38}

 [**Example 3.88**](#Example-3.88)\Pageref{Example-3.88}

 [**Example 3.89**](#example-R-minus-p)\Pageref{example-R-minus-p}

 [**Example 3.90**](#example-Rn-minus-p)\Pageref{example-Rn-minus-p}

 [**Theorem 3.91**](#theorem-connected)\Pageref{theorem-connected}

 [**Proof**](#Proof*-3.39)\Pageref{Proof*-3.39}

 [**Corollary 3.92**](#corollary-invariant-connected)\Pageref{corollary-invariant-connected}

 [**Example 3.93**](#Example-3.93)\Pageref{Example-3.93}

 [**Example 3.94**](#Example-3.94)\Pageref{Example-3.94}

 [**Definition 3.95**](#interval): Interval\Pageref{interval}

 [**Theorem 3.96**](#theorem-interval-connected)\Pageref{theorem-interval-connected}

 [**Proof**](#Proof*-3.40)\Pageref{Proof*-3.40}

 [**Theorem 3.97**](#intermediate-value-theorem-1): Intermediate Value Theorem\Pageref{intermediate-value-theorem-1}

 [**Proof**](#Proof*-3.41)\Pageref{Proof*-3.41}

 [**Definition 3.98**](#path-connectedness-1): Path connectedness\Pageref{path-connectedness-1}

 [**Example 3.99**](#Example-3.99)\Pageref{Example-3.99}

 [**Theorem 3.100**](#theorem-path-connected)\Pageref{theorem-path-connected}

 [**Proof**](#Proof*-3.42)\Pageref{Proof*-3.42}

 [**Lemma 3.101**](#lemma-connected-open-closed)\Pageref{lemma-connected-open-closed}

 [**Proof**](#Proof*-3.43)\Pageref{Proof*-3.43}

 [**Proposition 3.102**](#proposition-topologist-curve): Topologist curve\Pageref{proposition-topologist-curve}

 [**Proof**](#Proof*-3.44)\Pageref{Proof*-3.44}

 [**Question 4.1**](#Question-4.1)\Pageref{Question-4.1}

 [**Definition 4.2**](#bilinear-form): Bilinear form\Pageref{bilinear-form}

 [**Notation**](#Notation*-4.1)\Pageref{Notation*-4.1}

 [**Remark 4.3**](#matrix-representation-for-bilinear-forms): Matrix representation for bilinear forms\Pageref{matrix-representation-for-bilinear-forms}

 [**Definition 4.4**](#quadratic-form): Quadratic form\Pageref{quadratic-form}

 [**Proposition 4.5**](#Proposition-4.5)\Pageref{Proposition-4.5}

 [**Definition 4.6**](#inner-product): Inner product\Pageref{inner-product}

 [**Example 4.7**](#Example-4.7)\Pageref{Example-4.7}

 [**Proposition 4.8**](#Proposition-4.8)\Pageref{Proposition-4.8}

 [**Definition 4.9**](#linear-map): Linear map\Pageref{linear-map}

 [**Remark 4.10**](#matrix-representation-of-linear-maps): Matrix representation of linear maps\Pageref{matrix-representation-of-linear-maps}

 [**Definition 4.11**](#eigenvalues-and-eigenvectors): Eigenvalues and eigenvectors\Pageref{eigenvalues-and-eigenvectors}

 [**Definition 4.12**](#self-adjoint-map): Self-adjoint map\Pageref{self-adjoint-map}

 [**Theorem 4.13**](#theorem-spectral): Spectral Theorem\Pageref{theorem-spectral}

 [**Definition 4.14**](#Definition-4.14)\Pageref{Definition-4.14}

 [**Remark 4.15**](#remark-self-adoint)\Pageref{remark-self-adoint}

 [**Definition 4.16**](#matrix-eigenvalues): Matrix eigenvalues\Pageref{matrix-eigenvalues}

 [**Remark 4.17**](#Remark-4.17)\Pageref{Remark-4.17}

 [**Remark 4.18**](#Remark-4.18)\Pageref{Remark-4.18}

 [**Theorem 4.19**](#theorem-spectral-matrix): Spectral Theorem for matrices\Pageref{theorem-spectral-matrix}

 [**Remark 4.20**](#Remark-4.20)\Pageref{Remark-4.20}

 [**Definition 4.21**](#euclidean-topology): Euclidean Topology\Pageref{euclidean-topology}

 [**Definition 4.22**](#open-sets): Open Sets\Pageref{open-sets}

 [**Definition 4.23**](#closed-sets-1): Closed Sets\Pageref{closed-sets-1}

 [**Example 4.24**](#Example-4.24)\Pageref{Example-4.24}

 [**Definition 4.25**](#subspace-topology-1): Subspace Topology\Pageref{subspace-topology-1}

 [**Definition 4.26**](#continuous-function-1): Continuous Function\Pageref{continuous-function-1}

 [**Remark 4.27**](#Remark-4.27)\Pageref{Remark-4.27}

 [**Definition 4.28**](#homeomorphism): Homeomorphism\Pageref{homeomorphism}

 [**Definition 4.29**](#differentiable-function): Differentiable Function\Pageref{differentiable-function}

 [**Definition 4.30**](#partial-derivative): Partial Derivative\Pageref{partial-derivative}

 [**Definition 4.31**](#jacobian-matrix): Jacobian Matrix\Pageref{jacobian-matrix}

 [**Definition 4.32**](#multi-index-notation): Multi-index notation\Pageref{multi-index-notation}

 [**Definition 4.33**](#smooth-function): Smooth Function\Pageref{smooth-function}

 [**Notation**](#gradient-and-partial-derivatives): Gradient and partial derivatives\Pageref{gradient-and-partial-derivatives}

 [**Example 4.34**](#Example-4.34)\Pageref{Example-4.34}

 [**Definition 4.35**](#diffeomorphism-1): Diffeomorphism\Pageref{diffeomorphism-1}

 [**Theorem 4.36**](#inverse-function-theorem): Inverse Function Theorem\Pageref{inverse-function-theorem}

 [**Warning**](#Warning*-4.3)\Pageref{Warning*-4.3}

 [**Proposition 4.37**](#Proposition-4.37)\Pageref{Proposition-4.37}

 [**Example 4.38**](#Example-4.38)\Pageref{Example-4.38}

 [**Definition 4.39**](#surface): Surface\Pageref{surface}

 [**Remark 4.40**](#Remark-4.40)\Pageref{Remark-4.40}

 [**Notation**](#Notation*-4.4)\Pageref{Notation*-4.4}

 [**Example 4.41**](#d-plane-in-mathbbr3): 2D Plane in $\mathbb{R}^3$\Pageref{d-plane-in-mathbbr3}

 [**Example 4.42**](#unit-cylinder): Unit cylinder\Pageref{unit-cylinder}

 [**Important**](#Important*-4.5)\Pageref{Important*-4.5}

 [**Example 4.43**](#graph-of-a-function): Graph of a function\Pageref{graph-of-a-function}

 [**Example 4.44**](#circular-cone): Circular cone\Pageref{circular-cone}

 [**Definition 4.45**](#regular-chart): Regular Chart\Pageref{regular-chart}

 [**Proposition 4.46**](#Proposition-4.46)\Pageref{Proposition-4.46}

 [**Proof**](#Proof*-4.6)\Pageref{Proof*-4.6}

 [**Definition 4.47**](#regular-surface): Regular surface\Pageref{regular-surface}

 [**Example 4.48**](#d-plane-in-mathbbr3-1): 2D Plane in $\mathbb{R}^3$\Pageref{d-plane-in-mathbbr3-1}

 [**Example 4.49**](#example-unit-cylinder): Unit cylinder\Pageref{example-unit-cylinder}

 [**Example 4.50**](#graph-of-a-function-1): Graph of a function\Pageref{graph-of-a-function-1}

 [**Example 4.51**](#unit-sphere): Unit sphere\Pageref{unit-sphere}

 [**Remark 4.52**](#spherical-coordinates): Spherical coordinates\Pageref{spherical-coordinates}

 [**Example 4.53**](#unit-sphere-in-spherical-coordinates): Unit sphere in spherical coordinates\Pageref{unit-sphere-in-spherical-coordinates}

 [**Example 4.54**](#Example-4.54)\Pageref{Example-4.54}

 [**Definition 4.55**](#level-surface): Level surface\Pageref{level-surface}

 [**Theorem 4.56**](#theorem-level-surface)\Pageref{theorem-level-surface}

 [**Example 4.57**](#Example-4.57)\Pageref{Example-4.57}

 [**Example 4.58**](#circular-cone-1): Circular cone\Pageref{circular-cone-1}

 [**Definition 4.59**](#Definition-4.59)\Pageref{Definition-4.59}

 [**Remark 4.60**](#chain-rule): Chain rule\Pageref{chain-rule}

 [**Proposition 4.61**](#proposition-transition-maps)\Pageref{proposition-transition-maps}

 [**Proof**](#Proof*-4.7)\Pageref{Proof*-4.7}

 [**Definition 4.62**](#transition-map): Transition map\Pageref{transition-map}

 [**Theorem 4.63**](#theorem-transition-map)\Pageref{theorem-transition-map}

 [**Proposition 4.64**](#proposition-transition-map)\Pageref{proposition-transition-map}

 [**Proof**](#Proof*-4.8)\Pageref{Proof*-4.8}

 [**Important**](#Important*-4.9)\Pageref{Important*-4.9}

 [**Definition 4.65**](#definition-smooth-map-surface)\Pageref{definition-smooth-map-surface}

 [**Remark 4.66**](#Remark-4.66)\Pageref{Remark-4.66}

 [**Proposition 4.67**](#Proposition-4.67)\Pageref{Proposition-4.67}

 [**Proof**](#Proof*-4.10)\Pageref{Proof*-4.10}

 [**Definition 4.68**](#Definition-4.68)\Pageref{Definition-4.68}

 [**Proposition 4.69**](#proposition-f-chart)\Pageref{proposition-f-chart}

 [**Proof**](#Proof*-4.11)\Pageref{Proof*-4.11}

 [**Definition 4.70**](#local-diffeomorphism): Local diffeomorphism\Pageref{local-diffeomorphism}

 [**Assumption 4.71**](#Assumption-4.71)\Pageref{Assumption-4.71}

 [**Definition 4.72**](#tangent-vectors-and-tangent-space): Tangent vectors and tangent space\Pageref{tangent-vectors-and-tangent-space}

 [**Example 4.73**](#Example-4.73)\Pageref{Example-4.73}

 [**Lemma 4.74**](#lemma-curve-S)\Pageref{lemma-curve-S}

 [**Proof**](#Proof*-4.12)\Pageref{Proof*-4.12}

 [**Theorem 4.75**](#theorem-tangent-plane)\Pageref{theorem-tangent-plane}

 [**Proof**](#Proof*-4.13)\Pageref{Proof*-4.13}

 [**Definition 4.76**](#tangent-plane): Tangent plane\Pageref{tangent-plane}

 [**Remark 4.77**](#remark-affine-plane)\Pageref{remark-affine-plane}

 [**Proposition 4.78**](#equation-of-tangent-plane): Equation of tangent plane\Pageref{equation-of-tangent-plane}

 [**Proof**](#Proof*-4.14)\Pageref{Proof*-4.14}

 [**Example 4.79**](#Example-4.79)\Pageref{Example-4.79}

 [**Remark 4.80**](#tangent-space-and-derivations): Tangent space and derivations\Pageref{tangent-space-and-derivations}

 [**Lemma 4.81**](#lemma-derivative)\Pageref{lemma-derivative}

 [**Proof**](#Proof*-4.15)\Pageref{Proof*-4.15}

 [**Definition 4.82**](#differential-of-smooth-function): Differential of smooth function\Pageref{differential-of-smooth-function}

 [**Proposition 4.83**](#proposition-differential-smooth)\Pageref{proposition-differential-smooth}

 [**Proposition 4.84**](#Proposition-4.84)\Pageref{Proposition-4.84}

 [**Theorem 4.85**](#Theorem-4.85)\Pageref{Theorem-4.85}

 [**Definition 4.86**](#level-surface-1): Level surface\Pageref{level-surface-1}

 [**Theorem 4.87**](#theorem-level-surface-bis)\Pageref{theorem-level-surface-bis}

 [**Proposition 4.88**](#Proposition-4.88)\Pageref{Proposition-4.88}

 [**Proof**](#Proof*-4.16)\Pageref{Proof*-4.16}

 [**Theorem 4.89**](#theorem-quadrics)\Pageref{theorem-quadrics}

 [**Example 4.90**](#Example-4.90)\Pageref{Example-4.90}

 [**Example 4.91**](#Example-4.91)\Pageref{Example-4.91}

 [**Definition 4.92**](#ruled-surface): Ruled surface\Pageref{ruled-surface}

 [**Proposition 4.93**](#Proposition-4.93)\Pageref{Proposition-4.93}

 [**Proof**](#Proof*-4.17)\Pageref{Proof*-4.17}

 [**Example 4.94**](#unit-cylinder-1): Unit Cylinder\Pageref{unit-cylinder-1}

 [**Example 4.95**](#example-mobius): Möbius band\Pageref{example-mobius}

 [**Definition 4.96**](#surface-of-revolution): Surface of revolution\Pageref{surface-of-revolution}

 [**Proposition 4.97**](#proposition-revolution)\Pageref{proposition-revolution}

 [**Proof**](#Proof*-4.18)\Pageref{Proof*-4.18}

 [**Example 4.98**](#catenoid): Catenoid\Pageref{catenoid}

 [**Question 4.99**](#Question-4.99)\Pageref{Question-4.99}

 [**Definition 4.100**](#first-fundamental-form-1): First fundamental form\Pageref{first-fundamental-form-1}

 [**Definition 4.101**](#definition-coordinate-functions): Coordinate functions on tangent plane\Pageref{definition-coordinate-functions}

 [**Definition 4.102**](#first-fundamental-form-of-a-chart): First fundamental form of a chart\Pageref{first-fundamental-form-of-a-chart}

 [**Proposition 4.103**](#proposition-fff-quadratic)\Pageref{proposition-fff-quadratic}

 [**Proof**](#Proof*-4.19)\Pageref{Proof*-4.19}

 [**Remark 4.104**](#linear-algebra-interpretation): Linear algebra interpretation\Pageref{linear-algebra-interpretation}

 [**Notation**](#Notation*-4.20)\Pageref{Notation*-4.20}

 [**Remark 4.105**](#remark-fff-reparametrization): First fundamental form and reparametrizations\Pageref{remark-fff-reparametrization}

 [**Example 4.106**](#example-fff-plane): Plane\Pageref{example-fff-plane}

 [**Example 4.107**](#example-cylinder-fff): Unit cylinder\Pageref{example-cylinder-fff}

 [**Remark 4.108**](#Remark-4.108)\Pageref{Remark-4.108}

 [**Proposition 4.109**](#proposition-fff-length)\Pageref{proposition-fff-length}

 [**Proof**](#Proof*-4.21)\Pageref{Proof*-4.21}

 [**Example 4.110**](#cone): Cone\Pageref{cone}

 [**Definition 4.111**](#local-isometry): Local isometry\Pageref{local-isometry}

 [**Notation**](#Notation*-4.22)\Pageref{Notation*-4.22}

 [**Remark 4.112**](#remark-local-isometry)\Pageref{remark-local-isometry}

 [**Proposition 4.113**](#Proposition-4.113)\Pageref{Proposition-4.113}

 [**Proof**](#Proof*-4.23)\Pageref{Proof*-4.23}

 [**Theorem 4.114**](#theorem-local-isometry)\Pageref{theorem-local-isometry}

 [**Proof**](#Proof*-4.24)\Pageref{Proof*-4.24}

 [**Definition 4.115**](#angle-between-tangent-vectors): Angle between tangent vectors\Pageref{angle-between-tangent-vectors}

 [**Proposition 4.116**](#Proposition-4.116)\Pageref{Proposition-4.116}

 [**Proof**](#Proof*-4.25)\Pageref{Proof*-4.25}

 [**Definition 4.117**](#angle-between-curves-1): Angle between curves\Pageref{angle-between-curves-1}

 [**Proposition 4.118**](#Proposition-4.118)\Pageref{Proposition-4.118}

 [**Proof**](#Proof*-4.26)\Pageref{Proof*-4.26}

 [**Definition 4.119**](#conformal-map): Conformal map\Pageref{conformal-map}

 [**Remark 4.120**](#Remark-4.120)\Pageref{Remark-4.120}

 [**Proposition 4.121**](#Proposition-4.121)\Pageref{Proposition-4.121}

 [**Proof**](#Proof*-4.27)\Pageref{Proof*-4.27}

 [**Theorem 4.122**](#theorem-conformal)\Pageref{theorem-conformal}

 [**Proof**](#Proof*-4.28)\Pageref{Proof*-4.28}

 [**Corollary 4.123**](#corollary-conformal)\Pageref{corollary-conformal}

 [**Example 4.124**](#example-conformal): Conformal maps are not local isometries\Pageref{example-conformal}

 [**Definition 4.125**](#definition-conformally-flat): Conformally flat surface and conformal parametrization\Pageref{definition-conformally-flat}

 [**Theorem 4.126**](#Theorem-4.126)\Pageref{Theorem-4.126}

 [**Proof**](#Proof*-4.29)\Pageref{Proof*-4.29}

 [**Example 4.127**](#unit-cylinder-2): Unit cylinder\Pageref{unit-cylinder-2}

 [**Example 4.128**](#shpere): Shpere\Pageref{shpere}

 [**Question 4.129**](#Question-4.129)\Pageref{Question-4.129}

 [**Remark 4.130**](#remark-normal-chart)\Pageref{remark-normal-chart}

 [**Definition 4.131**](#standard-unit-normal-of-a-chart): Standard unit normal of a chart\Pageref{standard-unit-normal-of-a-chart}

 [**Definition 4.132**](#charts-with-same-orientation): Charts with same orientation\Pageref{charts-with-same-orientation}

 [**Example 4.133**](#Example-4.133)\Pageref{Example-4.133}

 [**Definition 4.134**](#orientable-surface): Orientable surface\Pageref{orientable-surface}

 [**Example 4.135**](#Example-4.135)\Pageref{Example-4.135}

 [**Example 4.136**](#Example-4.136)\Pageref{Example-4.136}

 [**Warning**](#orientability-is-a-global-property): Orientability is a global property\Pageref{orientability-is-a-global-property}

 [**Remark 4.137**](#Remark-4.137)\Pageref{Remark-4.137}

 [**Definition 4.138**](#unit-normal-of-a-surface): Unit normal of a surface\Pageref{unit-normal-of-a-surface}

 [**Warning**](#Warning*-4.31)\Pageref{Warning*-4.31}

 [**Proposition 4.139**](#Proposition-4.139)\Pageref{Proposition-4.139}

 [**Definition 4.140**](#standard-unit-normal-of-a-surface): Standard unit normal of a surface\Pageref{standard-unit-normal-of-a-surface}

 [**Notation**](#Notation*-4.32)\Pageref{Notation*-4.32}

 [**Definition 4.141**](#second-fundamental-form-of-a-chart): Second fundamental form of a chart\Pageref{second-fundamental-form-of-a-chart}

 [**Notation**](#Notation*-4.33)\Pageref{Notation*-4.33}

 [**Remark 4.142**](#second-fundamental-form-and-reparametrizations): Second fundamental form and reparametrizations\Pageref{second-fundamental-form-and-reparametrizations}

 [**Example 4.143**](#example-sff-plane): Plane\Pageref{example-sff-plane}

 [**Example 4.144**](#example-cylinder-sff): Unit cylinder\Pageref{example-cylinder-sff}

 [**Remark 4.145**](#Remark-4.145)\Pageref{Remark-4.145}

 [**Remark 4.146**](#Remark-4.146)\Pageref{Remark-4.146}

 [**Definition 4.147**](#gauss-map): Gauss map\Pageref{gauss-map}

 [**Remark 4.148**](#remark-gauss-coordinates)\Pageref{remark-gauss-coordinates}

 [**Example 4.149**](#Example-4.149)\Pageref{Example-4.149}

 [**Remark 4.150**](#Remark-4.150)\Pageref{Remark-4.150}

 [**Definition 4.151**](#weingarten-map): Weingarten map\Pageref{weingarten-map}

 [**Important**](#Important*-4.34)\Pageref{Important*-4.34}

 [**Remark 4.152**](#Remark-4.152)\Pageref{Remark-4.152}

 [**Definition 4.153**](#second-fundamental-form-of-a-surface): Second fundamental form of a surface\Pageref{second-fundamental-form-of-a-surface}

 [**Remark 4.154**](#Remark-4.154)\Pageref{Remark-4.154}

 [**Remark 4.155**](#matrix-of-the-second-fundamental-form): Matrix of the second fundamental form\Pageref{matrix-of-the-second-fundamental-form}

 [**Theorem 4.156**](#theorem-sff-chart)\Pageref{theorem-sff-chart}

 [**Lemma 4.157**](#lemma-normal-derivatives)\Pageref{lemma-normal-derivatives}

 [**Proof**](#Proof*-4.35)\Pageref{Proof*-4.35}

 [**Lemma 4.158**](#lemma-weingarten)\Pageref{lemma-weingarten}

 [**Proof**](#Proof*-4.36)\Pageref{Proof*-4.36}

 [**Proof**](#proof-of-theorem): Proof of Theorem 4.156\Pageref{proof-of-theorem}

 [**Theorem 4.159**](#theorem-weingarten): Matrix of Weingarten map\Pageref{theorem-weingarten}

 [**Proof**](#Proof*-4.38)\Pageref{Proof*-4.38}

 [**Important**](#Important*-4.39)\Pageref{Important*-4.39}

 [**Notation**](#Notation*-4.40)\Pageref{Notation*-4.40}

 [**Example 4.160**](#helicoid): Helicoid\Pageref{helicoid}

 [**Example 4.161**](#example-weingarten-2)\Pageref{example-weingarten-2}

 [**Definition 4.162**](#gaussian-and-mean-curvature-1): Gaussian and mean curvature\Pageref{gaussian-and-mean-curvature-1}

 [**Notation**](#trace-of-a-2-times-2-matrix): Trace of a $2 \times 2$ matrix\Pageref{trace-of-a-2-times-2-matrix}

 [**Remark 4.163**](#Remark-4.163)\Pageref{Remark-4.163}

 [**Proposition 4.164**](#Proposition-4.164)\Pageref{Proposition-4.164}

 [**Proof**](#Proof*-4.42)\Pageref{Proof*-4.42}

 [**Example 4.165**](#plane): Plane\Pageref{plane}

 [**Example 4.166**](#example-cylinder-curvatures): Unit cylinder\Pageref{example-cylinder-curvatures}

 [**Theorem 4.167**](#theorem-weingarten-diagonal)\Pageref{theorem-weingarten-diagonal}

 [**Proof**](#Proof*-4.43)\Pageref{Proof*-4.43}

 [**Corollary 4.168**](#corollary-weingarten-diagonal)\Pageref{corollary-weingarten-diagonal}

 [**Proof**](#Proof*-4.44)\Pageref{Proof*-4.44}

 [**Definition 4.169**](#principal-curvatures-and-vectors): Principal curvatures and vectors\Pageref{principal-curvatures-and-vectors}

 [**Remark 4.170**](#computing-principal-curvatures-and-vectors): Computing principal curvatures and vectors\Pageref{computing-principal-curvatures-and-vectors}

 [**Remark 4.171**](#computing-principal-curvatures-and-vectors-1): Computing principal curvatures and vectors\Pageref{computing-principal-curvatures-and-vectors-1}

 [**Proposition 4.172**](#Proposition-4.172)\Pageref{Proposition-4.172}

 [**Proof**](#Proof*-4.45)\Pageref{Proof*-4.45}

 [**Important**](#Important*-4.46)\Pageref{Important*-4.46}

 [**Example 4.173**](#unit-cylinder-3): Unit Cylinder\Pageref{unit-cylinder-3}

 [**Example 4.174**](#example-sphere-principal): Sphere\Pageref{example-sphere-principal}

 [**Example 4.175**](#torus): Torus\Pageref{torus}

 [**Question 4.176**](#Question-4.176)\Pageref{Question-4.176}

 [**Proposition 4.177**](#proposition-surface-frame)\Pageref{proposition-surface-frame}

 [**Proof**](#Proof*-4.47)\Pageref{Proof*-4.47}

 [**Important**](#Important*-4.48)\Pageref{Important*-4.48}

 [**Proposition 4.178**](#Proposition-4.178)\Pageref{Proposition-4.178}

 [**Proof**](#Proof*-4.49)\Pageref{Proof*-4.49}

 [**Definition 4.179**](#normal-and-geodesic-curvature): Normal and geodesic curvature\Pageref{normal-and-geodesic-curvature}

 [**Theorem 4.180**](#theorem-normal-curvature)\Pageref{theorem-normal-curvature}

 [**Proof**](#Proof*-4.50)\Pageref{Proof*-4.50}

 [**Example 4.181**](#example-curves-on-sphere): Curves on the sphere\Pageref{example-curves-on-sphere}

 [**Theorem 4.182**](#theorem-euler-thm): Euler’s Theorem\Pageref{theorem-euler-thm}

 [**Proof**](#Proof*-4.51)\Pageref{Proof*-4.51}

 [**Corollary 4.183**](#corollary-euler-thm)\Pageref{corollary-euler-thm}

 [**Example 4.184**](#curves-on-the-sphere): Curves on the sphere\Pageref{curves-on-the-sphere}

 [**Theorem 4.185**](#theorem-local-surface): Local structure of surfaces\Pageref{theorem-local-surface}

 [**Proof**](#Proof*-4.52)\Pageref{Proof*-4.52}

 [**Definition 4.186**](#local-shape-types): Local shape types\Pageref{local-shape-types}

 [**Example 4.187**](#Example-4.187)\Pageref{Example-4.187}

 [**Definition 4.188**](#umbilical-point): Umbilical point\Pageref{umbilical-point}

 [**Remark 4.189**](#Remark-4.189)\Pageref{Remark-4.189}

 [**Theorem 4.190**](#Theorem-4.190)\Pageref{Theorem-4.190}

 [**Proof**](#Proof*-4.53)\Pageref{Proof*-4.53}