The first problem in multivariable analysis is that points in $\mathbb{R}^n$ are not arranged on a line. In $\mathbb{R}$, the distance between $x$ and $y$ is $|x-y|$, and limits, continuity, open intervals, derivatives, and compactness all lean on that single quantity. In $\mathbb{R}^n$, a displacement has several components, so we need a way to turn the coordinate differences into one nonnegative number that behaves like distance. The Euclidean metric is the standard answer. It measures the length of the straight-line displacement from $x$ to $y$ by taking the square root of the sum of the squared coordinate differences. This is the metric secretly used whenever we draw circles in the plane, balls in space, or small neighbourhoods in a coordinate chart. A bad distance formula breaks analysis quickly. If we tried to measure displacement in $\mathbb{R}^2$ by adding coordinates with signs, the distance from $(0,0)$ to $(1,-1)$ would be $0$, even though the points are different. If we ignored one coordinate, sequences could move far away in the ignored direction while appearing to converge. The Euclidean metric avoids these failures by measuring every coordinate symmetrically and by making the [Pythagorean theorem](/theorems/3266) part of the geometry. [example: A Failed Coordinate Sum] Define $s:\mathbb{R}^2\times\mathbb{R}^2\to\mathbb{R}$ by \begin{align*} s(x,y)=(x_1-y_1)+(x_2-y_2) \end{align*} for $x=(x_1,x_2)$ and $y=(y_1,y_2)$. Take $x=(0,0)$ and $y=(1,-1)$. These points are distinct because their first coordinates are different: $0\ne1$. But the coordinate-sum formula gives \begin{align*} s((0,0),(1,-1))=(0-1)+(0-(-1)) \end{align*} and hence \begin{align*} s((0,0),(1,-1))=-1+1=0. \end{align*} Therefore $s$ assigns value $0$ to two distinct points, so it cannot satisfy the separation requirement for a metric. Replacing $s$ by $|s|$ does not repair the failure, because \begin{align*} |s((0,0),(1,-1))|=|0|=0. \end{align*} The problem is cancellation: the displacement from $x$ to $y$ is \begin{align*} x-y=(0-1,0-(-1))=(-1,1), \end{align*} and the signed aggregate $(-1)+1$ loses the fact that both coordinates changed. A genuine distance must control the whole displacement vector rather than only the sum of its signed coordinate differences. [/example] The rest of the chapter builds the Euclidean metric from the Euclidean norm, checks that it satisfies the metric axioms, and explains why this single formula controls the familiar topology, convergence, compactness, and calculus of $\mathbb{R}^n$. ## Definition The page topic is the distance function itself, so we define it before abstracting its ingredients. The formula should assign zero distance only to equal points, should be unchanged when the two points are swapped, and should measure all coordinate differences without allowing positive and negative coordinates to cancel. Squaring the coordinate differences is the simplest way to enforce that behaviour while preserving the Pythagorean geometry of straight-line displacement. [definition: Euclidean Metric] For $n\in\mathbb{N}$, the Euclidean metric on $\mathbb{R}^n$ is the function $d_2:\mathbb{R}^n\times\mathbb{R}^n\to[0,\infty)$ defined by \begin{align*} d_2(x,y) = \left(\sum_{k=1}^n |x_k-y_k|^2\right)^{1/2} \end{align*} for $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$. [/definition] The notation $d_2$ records that the formula uses squares and a square root. In most analysis on $\mathbb{R}^n$, this formula is so standard that authors write $|x-y|$ instead of $d_2(x,y)$. This shorthand only works because there is already a notion of the length of a single displacement vector. Before using $|x-y|$ freely, we need to name that length function and distinguish it from the two-point distance it generates. [definition: Euclidean Norm] For $n\in\mathbb{N}$, the Euclidean norm is the function $|\cdot|:\mathbb{R}^n\to[0,\infty)$ defined by \begin{align*} |x| = \left(\sum_{k=1}^n |x_k|^2\right)^{1/2} \end{align*} for $x=(x_1,\dots,x_n)$. [/definition] The square root is not decoration: it makes $|x|^2$ additive across perpendicular coordinate directions. That is the algebraic form of the Pythagorean theorem, and it is why spheres, rotations, and orthogonal projections fit naturally with this norm. The norm gives lengths of vectors, while the Euclidean metric applies that length formula to the displacement vector $x-y$. The problem now is certification: before this formula can support limits and open sets, we need an abstract checklist for what any distance function must satisfy. That checklist is the metric definition. [definition: Metric] Let $X$ be a set. A metric on $X$ is a function $d:X\times X\to[0,\infty)$ such that for all $x,y,z\in X$, \begin{align*} d(x,y)=0 \iff x=y, \end{align*} \begin{align*} d(x,y)=d(y,x), \end{align*} and \begin{align*} d(x,z)\le d(x,y)+d(y,z). \end{align*} [/definition] The first condition says distance separates points. The second says distance has no direction. The third, the triangle inequality, says a direct path is no longer than a path through an intermediate point. For the Euclidean formula, the separation and symmetry properties follow directly from the coordinate expression; the serious estimate is the triangle inequality. For vectors $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$ in $\mathbb{R}^n$, write \begin{align*} x\cdot y=\sum_{k=1}^n x_k y_k \end{align*} for the Euclidean dot product, and write $|x|=\sqrt{x\cdot x}$ for Euclidean length. The estimate needed here is the coordinate Cauchy--Schwarz inequality: \begin{align*} |x\cdot y|\le |x|\,|y|. \end{align*} It says that the dot product cannot exceed the product of the two lengths. This elementary form is enough to prove $|x+y|\le |x|+|y|$, so the coordinate formula becomes a genuine metric rather than just a plausible distance candidate. [quotetheorem:8512] This theorem is the permission slip for using every construction from [metric spaces](/page/Metric%20Space) in $\mathbb{R}^n$. Once $d_2$ is known to be a metric, open balls, closed sets, limits, continuity, and compactness can be defined without returning to coordinates every time. ## Balls and the Euclidean Topology ### Open and Closed Balls The metric becomes useful when it tells us what it means for points to be near each other. The basic neighbourhoods are balls: all points whose distance from a center is less than a chosen radius. In the Euclidean metric, these are the usual intervals, disks, and solid balls, so they connect the formula for distance to the visible geometry of neighbourhoods. [definition: Euclidean Open Ball] Let $x_0\in\mathbb{R}^n$ and let $r>0$. The Euclidean open ball with center $x_0$ and radius $r$ is \begin{align*} B(x_0,r) = \{x\in\mathbb{R}^n : d_2(x,x_0)<r\}. \end{align*} [/definition] The strict inequality makes the boundary points excluded. In $\mathbb{R}$, $B(x_0,r)$ is the interval $(x_0-r,x_0+r)$. In $\mathbb{R}^2$, it is the open disk of radius $r$. Many arguments also need the version that includes limiting boundary points, because maxima, minima, and subsequential limits often occur on the boundary. [definition: Euclidean Closed Ball] Let $x_0\in\mathbb{R}^n$ and let $r\ge 0$. The Euclidean closed ball with center $x_0$ and radius $r$ is \begin{align*} \overline{B}(x_0,r) = \{x\in\mathbb{R}^n : d_2(x,x_0)\le r\}. \end{align*} [/definition] Closed balls are the simplest bounded closed sets in Euclidean space. Their importance comes from the fact that a set is bounded exactly when it fits inside some closed ball centered at the origin. The next problem is larger than a single ball: analysis needs a rule for which arbitrary subsets have enough local room around every point. That rule is the Euclidean topology. [definition: Euclidean Topology] The Euclidean topology on $\mathbb{R}^n$ is the collection of subsets $U\subset\mathbb{R}^n$ such that for every $x\in U$ there exists $r>0$ with \begin{align*} B(x,r)\subset U. \end{align*} [/definition] This definition turns local room into the central test for openness. A point belongs to an open set only if it can move a positive Euclidean distance in every sufficiently small direction and remain in the set. The following example grounds the definition in the familiar low-dimensional pictures before we use it abstractly. [example: Open Balls in $\mathbb{R}$ and $\mathbb{R}^2$] For $r>0$, the open ball in $\mathbb{R}$ with center $x_0$ is defined by the condition $d_2(x,x_0)<r$. Since the one-dimensional Euclidean metric is \begin{align*} d_2(x,x_0)=|x-x_0|, \end{align*} we have \begin{align*} x\in B(x_0,r) \iff |x-x_0|<r. \end{align*} The absolute value inequality is equivalent to \begin{align*} -r<x-x_0<r. \end{align*} Adding $x_0$ to all three parts gives \begin{align*} x_0-r<x<x_0+r. \end{align*} Therefore \begin{align*} B(x_0,r)=(x_0-r,x_0+r). \end{align*} In $\mathbb{R}^2$, let $x=(x_1,x_2)$. The open ball centered at $0=(0,0)$ with radius $1$ consists of the points satisfying $d_2(x,0)<1$. By the Euclidean metric formula, \begin{align*} d_2(x,0)=\left(|x_1-0|^2+|x_2-0|^2\right)^{1/2}. \end{align*} Since $|x_1-0|^2=x_1^2$ and $|x_2-0|^2=x_2^2$, this becomes \begin{align*} d_2(x,0)=\left(x_1^2+x_2^2\right)^{1/2}. \end{align*} Thus \begin{align*} x\in B(0,1) \iff \left(x_1^2+x_2^2\right)^{1/2}<1. \end{align*} Both sides are nonnegative, so squaring preserves the inequality: \begin{align*} \left(x_1^2+x_2^2\right)^{1/2}<1 \iff x_1^2+x_2^2<1. \end{align*} Hence \begin{align*} B(0,1)=\{(x_1,x_2)\in\mathbb{R}^2:x_1^2+x_2^2<1\}. \end{align*} The same Euclidean formula therefore gives open intervals in one dimension and open disks in two dimensions. [/example] ### Balls as a Basis The ball definition is local, but it is strong enough to generate the full topology. This matters because proofs about open sets should reduce to proofs about sufficiently small balls around individual points. The precise basis statement guarantees that the intersection of two local neighbourhoods still contains a smaller neighbourhood of the same form. [quotetheorem:7784] This theorem explains why local estimates are enough in Euclidean analysis. To prove continuity, openness, or local boundedness, it often suffices to solve the problem on a small ball around each point. ## Convergence and Continuity ### Limits of Sequences A metric turns convergence into a numerical statement: distances to the proposed limit must tend to zero. In $\mathbb{R}^n$, this definition must also agree with the coordinate-by-coordinate way we compute limits. If it did not, the metric would conflict with the algebraic structure of Euclidean space. [definition: Convergence in the Euclidean Metric] A sequence $(x_j)_{j=1}^{\infty}$ in $\mathbb{R}^n$ converges to $x\in\mathbb{R}^n$ in the Euclidean metric if \begin{align*} d_2(x_j,x) \to 0 \end{align*} as $j\to\infty$. [/definition] This definition compresses $n$ coordinate limits into one scalar limit. The compression loses no information because the Euclidean norm controls every coordinate difference and is controlled by the sum of all coordinate differences. Explicitly, for $x_j=(x_{j,1},\dots,x_{j,n})$ and $x=(x_1,\dots,x_n)$, \begin{align*} x_j\to x \text{ in the Euclidean metric} \end{align*} if and only if \begin{align*} x_{j,k}\to x_k \end{align*} for each coordinate $k=1,\dots,n$. One direction follows from $|x_{j,k}-x_k|\le d_2(x_j,x)$; the other follows because a finite sum of coordinate errors tending to zero also tends to zero. This equivalence is why multivariable limit calculations can often be checked component by component. It also explains a common warning: convergence in $\mathbb{R}^n$ is not a new kind of limiting process; it is coordinate convergence organized by a geometry. ### Continuity and Approach Paths Continuity should mean that nearby inputs produce nearby outputs. In Euclidean spaces, both uses of “nearby” are measured by Euclidean balls, possibly in different dimensions. This gives the familiar $\varepsilon$-$\delta$ definition in a form that works for maps $\mathbb{R}^n\to\mathbb{R}^m$. [definition: Euclidean Continuity] Let $U\subset\mathbb{R}^n$, let $m\in\mathbb{N}$, and let $f:U\to\mathbb{R}^m$ be a function. The function $f$ is continuous at $x_0\in U$ with respect to the Euclidean metrics if for every $\varepsilon>0$ there exists $\delta>0$ such that for all $x\in U$, \begin{align*} d_2(x,x_0)<\delta \implies d_2(f(x),f(x_0))<\varepsilon. \end{align*} [/definition] The definition has the same shape as in single-variable analysis, but now the input condition describes a ball around $x_0$ and the output condition describes a ball around $f(x_0)$. This is the form that survives in metric spaces, normed spaces, and manifolds after choosing charts. It also explains why checking only one path of approach is not enough to prove a multivariable limit. [example: A Path Test That Detects Non-Convergence] Consider $f:\mathbb{R}^2\setminus\{0\}\to\mathbb{R}$ defined by \begin{align*} f(x_1,x_2)=\frac{x_1x_2}{x_1^2+x_2^2}. \end{align*} We show that $f$ has no Euclidean limit at $(0,0)$ by comparing two ways of approaching the origin. On the coordinate axis $x_2=0$, for every $t\ne0$, \begin{align*} f(t,0)=\frac{t\cdot0}{t^2+0^2}. \end{align*} Since $t^2+0^2=t^2$ and $t\ne0$, this gives \begin{align*} f(t,0)=\frac{0}{t^2}=0. \end{align*} The Euclidean distance from $(t,0)$ to $(0,0)$ is \begin{align*} d_2((t,0),(0,0))=\left(|t-0|^2+|0-0|^2\right)^{1/2}. \end{align*} Thus \begin{align*} d_2((t,0),(0,0))=\left(|t|^2+0\right)^{1/2}=|t|, \end{align*} so $(t,0)\to(0,0)$ as $t\to0$. On the diagonal path $x_2=x_1$, for every $t\ne0$, \begin{align*} f(t,t)=\frac{t\cdot t}{t^2+t^2}. \end{align*} Since $t^2+t^2=2t^2$ and $t\ne0$, this gives \begin{align*} f(t,t)=\frac{t^2}{2t^2}=\frac12. \end{align*} The Euclidean distance from $(t,t)$ to $(0,0)$ is \begin{align*} d_2((t,t),(0,0))=\left(|t-0|^2+|t-0|^2\right)^{1/2}. \end{align*} Hence \begin{align*} d_2((t,t),(0,0))=\left(2|t|^2\right)^{1/2}=\sqrt{2}\,|t|, \end{align*} so $(t,t)\to(0,0)$ as $t\to0$. If $f$ had a Euclidean limit $L$ at $(0,0)$, then for $\varepsilon=1/8$ there would be $\delta>0$ such that $d_2(x,(0,0))<\delta$ implies $|f(x)-L|<1/8$. Choose $t\ne0$ with $|t|<\delta/\sqrt{2}$. Then both $(t,0)$ and $(t,t)$ are within Euclidean distance $\delta$ of $(0,0)$, so \begin{align*} |0-L|<\frac18 \end{align*} and \begin{align*} \left|\frac12-L\right|<\frac18. \end{align*} By the triangle inequality in $\mathbb{R}$, \begin{align*} \frac12=\left|\frac12-0\right|\le\left|\frac12-L\right|+|L-0|. \end{align*} The two displayed inequalities give \begin{align*} \left|\frac12-L\right|+|L-0|<\frac18+\frac18=\frac14, \end{align*} contradicting $\frac12\le\frac14$. Therefore $f$ has no Euclidean limit at $(0,0)$; approaching the same point along different Euclidean paths forces incompatible limiting values. [/example] This example shows why the Euclidean metric is stronger than checking a single line or coordinate axis. To have a limit at a point, the function must behave consistently along every way of approaching the point within small Euclidean balls. ## Inequalities and Comparison of Distances ### Triangle-Type Estimates The Euclidean metric is geometrically natural, but proofs often require estimates rather than pictures. The triangle inequality controls sums of displacements, while the [reverse triangle inequality](/theorems/2300) controls changes in lengths. These estimates are the standard tools for turning geometric nearness into algebraic inequalities. For vectors $x,y\in\mathbb{R}^n$, the Euclidean triangle inequality is \begin{align*} |x+y|\le |x|+|y|. \end{align*} In metric language, substituting $x=a-b$ and $y=b-c$ gives \begin{align*} d_2(a,c)\le d_2(a,b)+d_2(b,c). \end{align*} This is the algebraic expression of the idea that detours do not shorten straight-line distance. The companion estimate is the Euclidean reverse triangle inequality: \begin{align*} \bigl||x|-|y|\bigr|\le |x-y|. \end{align*} It follows by applying the triangle inequality twice, once to $x=(x-y)+y$ and once to $y=(y-x)+x$. Thus the length function $x\mapsto |x|$ is continuous and even $1$-Lipschitz: a small change in the vector can cause at most the same small change in its length. ### Coordinate Boxes and Round Balls Many arguments compare the Euclidean metric with simpler coordinate bounds. Rectangular boxes are often easier to describe than balls, especially when estimating each coordinate separately. To make this comparison precise, we introduce the norm that records the largest coordinate size. [definition: Supremum Norm on $\mathbb{R}^n$] The supremum norm is the function $|\cdot|_\infty:\mathbb{R}^n\to[0,\infty)$ defined by \begin{align*} |x|_\infty = \max_{1\le k\le n}|x_k| \end{align*} for $x=(x_1,\dots,x_n)$. [/definition] The supremum norm measures the largest coordinate size. Its balls are boxes rather than round balls, so it is often better suited for coordinate estimates and product constructions. The essential question is whether replacing round balls by boxes changes limits and open sets; the comparison theorem says it does not in fixed finite dimension. [quotetheorem:8513] This comparison shows that Euclidean balls and coordinate boxes define the same notion of nearness. The constants depend on $n$, which is harmless in fixed finite dimension but becomes important when dimension varies. [example: A Euclidean Ball Sits Between Two Boxes] Let $r>0$. In $\mathbb{R}^2$, write $x=(x_1,x_2)$, so \begin{align*} |x|=\left(x_1^2+x_2^2\right)^{1/2} \end{align*} and \begin{align*} |x|_\infty=\max\{|x_1|,|x_2|\}. \end{align*} First suppose $|x|_\infty<r/\sqrt{2}$. Then $|x_1|<r/\sqrt{2}$ and $|x_2|<r/\sqrt{2}$, so \begin{align*} x_1^2<\frac{r^2}{2} \end{align*} and \begin{align*} x_2^2<\frac{r^2}{2}. \end{align*} Adding these inequalities gives \begin{align*} x_1^2+x_2^2<r^2. \end{align*} Since both sides are nonnegative and the square-root function is increasing on $[0,\infty)$, \begin{align*} |x|=\left(x_1^2+x_2^2\right)^{1/2}<r. \end{align*} Thus $x\in B(0,r)$, proving \begin{align*} \{x\in\mathbb{R}^2: |x|_\infty<r/\sqrt{2}\}\subset B(0,r). \end{align*} Conversely, suppose $x\in B(0,r)$. Then $|x|<r$. Because $x_1^2\le x_1^2+x_2^2$ and $x_2^2\le x_1^2+x_2^2$, taking square roots gives \begin{align*} |x_1|\le |x| \end{align*} and \begin{align*} |x_2|\le |x|. \end{align*} Therefore \begin{align*} |x|_\infty=\max\{|x_1|,|x_2|\}\le |x|<r. \end{align*} Hence \begin{align*} B(0,r)\subset \{x\in\mathbb{R}^2: |x|_\infty<r\}. \end{align*} Combining the two inclusions, \begin{align*} \{x\in\mathbb{R}^2: |x|_\infty<r/\sqrt{2}\} \subset B(0,r) \subset \{x\in\mathbb{R}^2: |x|_\infty<r\}. \end{align*} So every Euclidean disk contains a smaller coordinate square and is contained in a larger coordinate square, which is the local comparison behind the agreement of rectangular and round neighbourhoods. [/example] The finite-dimensional [comparison principle](/theorems/4870) is a recurring theme. In finite-dimensional Euclidean space, many reasonable ways to measure vector size agree topologically, even when their unit balls have different shapes. ## Compactness and Completeness ### Cauchy Sequences The Euclidean metric also decides which limiting processes stay inside a set. Completeness concerns sequences whose late terms become mutually close, even before a limit has been identified. This property is essential for iterative arguments, because many constructions first prove that approximations are Cauchy and only later identify the object they approach. [definition: Euclidean Cauchy Sequence] A sequence $(x_j)_{j=1}^{\infty}$ in $\mathbb{R}^n$ is Cauchy in the Euclidean metric if for every $\varepsilon>0$ there exists $N\in\mathbb{N}$ such that for all $j,k\ge N$, \begin{align*} d_2(x_j,x_k)<\varepsilon. \end{align*} [/definition] A [Cauchy sequence](/page/Cauchy%20Sequence) is internally convergent-looking: its late terms become close to one another. The missing question is whether there is an actual point of the space that the sequence approaches. In Euclidean space, the answer is yes because the coordinate sequences are Cauchy in $\mathbb{R}$. [quotetheorem:8514] Completeness is why iterative constructions in analysis can converge to actual Euclidean points. It is inherited coordinate by coordinate from the completeness of $\mathbb{R}$. ### Closed and Bounded Sets Completeness alone does not stop sequences from escaping to infinity. Compactness needs a second ingredient: boundedness, which places the entire set inside one large Euclidean ball. This is the metric condition that rules out escape in arbitrarily large directions. [definition: Euclidean Bounded Set] A set $A\subset\mathbb{R}^n$ is bounded in the Euclidean metric if there exists $R>0$ such that \begin{align*} A\subset B(0,R). \end{align*} [/definition] Boundedness alone is not compactness: the open interval $(0,1)$ is bounded but misses its endpoint limits. Closedness supplies the missing boundary points. The central [compactness theorem](/theorems/2748) for Euclidean space says these two concrete conditions are exactly what compactness requires. [quotetheorem:315] The Heine--Borel theorem is one of the main rewards of the Euclidean metric. It converts the abstract open-cover definition of compactness into two concrete geometric tests: no missing boundary points, and no escape to infinity. [example: Bounded But Not Compact] The set $A=(0,1)\subset\mathbb{R}$ is bounded in the Euclidean metric. Indeed, if $x\in A$, then $0<x<1$, so \begin{align*} d_2(x,0)=|x-0|=|x|=x<1<2. \end{align*} Therefore $x\in B(0,2)$ for every $x\in A$, and hence \begin{align*} A\subset B(0,2). \end{align*} To see that $A$ is not compact, consider the family of Euclidean open intervals \begin{align*} U_m=\left(\frac1m,1\right) \end{align*} for integers $m\ge2$. If $x\in(0,1)$, then $1/x>1$, so by the [Archimedean property](/theorems/737) we may choose an integer $m\ge2$ with $m>1/x$. Since $x>0$, this inequality gives \begin{align*} \frac1m<x. \end{align*} Together with $x<1$, this shows $x\in U_m$. Hence \begin{align*} A\subset \bigcup_{m=2}^{\infty}U_m. \end{align*} The reverse inclusion is immediate from $U_m\subset(0,1)$, so the sets $U_m$ form an [open cover](/page/Open%20Cover) of $A$. No finite subcollection covers $A$. If $U_{m_1},\dots,U_{m_N}$ are chosen and $M=\max\{m_1,\dots,m_N\}$, then for each $i$ we have $m_i\le M$, so \begin{align*} \frac1{m_i}\ge \frac1M. \end{align*} Thus \begin{align*} U_{m_i}=\left(\frac1{m_i},1\right)\subset \left(\frac1M,1\right) \end{align*} for every $i$, and therefore \begin{align*} \bigcup_{i=1}^N U_{m_i}\subset \left(\frac1M,1\right). \end{align*} But the point $1/(M+1)$ satisfies \begin{align*} 0<\frac1{M+1}<\frac1M<1, \end{align*} so $1/(M+1)\in A$ while $1/(M+1)\notin(1/M,1)$. Hence the finite subcollection does not cover $A$. Thus $A=(0,1)$ is bounded but not compact, showing that boundedness without closedness does not guarantee compactness. [/example] The previous example shows that bounded sets can still lose limiting points. Compact sets avoid both failures: they are bounded enough to prevent escape and closed enough to retain limits. This is exactly the setting needed when an optimization problem asks for an actual point where a best value is reached. [quotetheorem:304] This theorem is the analytic reason compact Euclidean sets matter. Optimization, variational problems, and many existence arguments begin by placing a problem inside a closed and bounded region. ## Geometry Preserved by the Metric ### Isometries The Euclidean metric is not merely any metric on $\mathbb{R}^n$. It is the metric compatible with translations, rotations, and reflections. These symmetries explain why Euclidean geometry can move figures around without changing lengths, and the correct abstraction is a distance-preserving map. [definition: Euclidean Isometry] Let $A\subset\mathbb{R}^n$ and $B\subset\mathbb{R}^m$. A function $F:A\to B$ is a Euclidean isometry if for all $x,y\in A$, \begin{align*} d_2(F(x),F(y)) = d_2(x,y). \end{align*} [/definition] An isometry preserves every distance, so it preserves balls, convergence, Cauchy sequences, and boundedness. Because it is continuous, it also sends compact subsets of its domain to compact subsets of its codomain. It is the correct formal expression of a rigid motion when the domain and codomain are Euclidean subsets. [example: Translations Preserve Euclidean Distance] Fix $a\in\mathbb{R}^n$ and define $T_a:\mathbb{R}^n\to\mathbb{R}^n$ by \begin{align*} T_a(x)=x+a. \end{align*} Let $x=(x_1,\dots,x_n)$, $y=(y_1,\dots,y_n)$, and $a=(a_1,\dots,a_n)$. Then \begin{align*} T_a(x)=(x_1+a_1,\dots,x_n+a_n) \end{align*} and \begin{align*} T_a(y)=(y_1+a_1,\dots,y_n+a_n). \end{align*} Therefore the $k$th coordinate difference is \begin{align*} (T_a(x))_k-(T_a(y))_k=(x_k+a_k)-(y_k+a_k). \end{align*} Associativity and commutativity of addition in $\mathbb{R}$ give \begin{align*} (x_k+a_k)-(y_k+a_k)=x_k+a_k-y_k-a_k=x_k-y_k. \end{align*} Using the Euclidean metric formula, \begin{align*} d_2(T_a(x),T_a(y))=\left(\sum_{k=1}^n |(T_a(x))_k-(T_a(y))_k|^2\right)^{1/2}. \end{align*} Substituting the coordinate identity above gives \begin{align*} d_2(T_a(x),T_a(y))=\left(\sum_{k=1}^n |x_k-y_k|^2\right)^{1/2}. \end{align*} By the Euclidean metric formula again, \begin{align*} \left(\sum_{k=1}^n |x_k-y_k|^2\right)^{1/2}=d_2(x,y). \end{align*} Hence \begin{align*} d_2(T_a(x),T_a(y))=d_2(x,y). \end{align*} Thus translating both points by the same vector cancels out of every coordinate difference, so every translation is a Euclidean isometry. [/example] ### Orthogonal Symmetry Translations explain why Euclidean distance does not depend on absolute position. The other basic rigid motions are rotations and reflections, which are linear maps preserving the Euclidean [inner product](/page/Inner%20Product). To state this algebraically, we need the class of matrices whose columns form an orthonormal coordinate system. Here $Q^\top$ denotes the transpose of $Q$, obtained by interchanging rows and columns, and $I_n$ denotes the $n\times n$ identity matrix. [definition: Orthogonal Matrix] A matrix $Q\in\mathbb{R}^{n\times n}$ is orthogonal if \begin{align*} Q^\top Q = I_n. \end{align*} [/definition] Orthogonal matrices encode the linear symmetries of Euclidean space. They may rotate, reflect, or combine both operations, but the key metric question is whether they preserve the distance between every pair of points. The defining equation $Q^\top Q=I_n$ is built to preserve lengths of displacement vectors, so it should turn each linear orthogonal map into a Euclidean isometry. In the theorem below, $M_n(\mathbb{R})$ denotes the set of all $n\times n$ real matrices, the same [matrix space](/page/Matrix%20Space) written above as $\mathbb{R}^{n\times n}$. [quotetheorem:8282] This theorem identifies the metric content of orthogonality. An orthogonal matrix can change coordinates, rotate a figure, or reflect it across a subspace, but it cannot stretch a displacement vector or change the distance between two points. For example, in the plane, the usual rotation matrices and reflection matrices across lines through the origin are orthogonal, while a shear or a nontrivial scaling is not. The limitation is just as important: orthogonal maps are the linear rigid motions fixing the origin, so translations must be handled separately when describing all Euclidean isometries. Together with [translation invariance](/theorems/4911), this result explains why Euclidean geometry can move figures around without changing their metric content, and it will underlie later uses of coordinates, derivative estimates, and local geometric models. ## Euclidean Metric in Analysis and Geometry ### Differentiability The Euclidean metric is the default local model for analysis. It gives the balls used in limits, the norms used in derivative estimates, and the local coordinate distances used in manifold charts. A differentiable function is well approximated by a [linear map](/page/Linear%20Map) when the Euclidean distance to the base point is small, so the phrase $h\to0$ in multivariable calculus means $|h|\to0$ in the Euclidean norm. [definition: Differentiability in Euclidean Space] Let $U\subset\mathbb{R}^n$ be open, let $m\in\mathbb{N}$, and let $f:U\to\mathbb{R}^m$ be a function. The function $f$ is differentiable at $a\in U$ if there exists a linear map $Df_a:\mathbb{R}^n\to\mathbb{R}^m$ such that \begin{align*} \frac{|f(a+h)-f(a)-Df_a(h)|}{|h|} \to 0 \end{align*} as $h\to0$ in $\mathbb{R}^n$, with $h\ne0$ and $a+h\in U$. [/definition] The Euclidean norm appears twice: it measures the error in the output and the size of the input displacement. Without the metric, the phrase “linear approximation at small scale” would have no quantitative meaning. [example: Differentiability Uses Euclidean Scale] Let $f:\mathbb{R}^2\to\mathbb{R}$ be defined by \begin{align*} f(x_1,x_2)=x_1^2+x_2^2. \end{align*} Fix $a=(a_1,a_2)$ and write $h=(h_1,h_2)$. We verify that the linear map \begin{align*} Df_a(h)=2a_1h_1+2a_2h_2 \end{align*} has an error term small compared with the Euclidean length of $h$. Since $a+h=(a_1+h_1,a_2+h_2)$, we have \begin{align*} f(a+h)=(a_1+h_1)^2+(a_2+h_2)^2. \end{align*} Expanding each square gives \begin{align*} f(a+h)=a_1^2+2a_1h_1+h_1^2+a_2^2+2a_2h_2+h_2^2. \end{align*} Also \begin{align*} f(a)=a_1^2+a_2^2. \end{align*} Therefore \begin{align*} f(a+h)-f(a)=2a_1h_1+2a_2h_2+h_1^2+h_2^2. \end{align*} Subtracting $Df_a(h)=2a_1h_1+2a_2h_2$ leaves \begin{align*} f(a+h)-f(a)-Df_a(h)=h_1^2+h_2^2. \end{align*} By the Euclidean norm formula in $\mathbb{R}^2$, \begin{align*} |h|=\left(h_1^2+h_2^2\right)^{1/2}, \end{align*} so \begin{align*} |h|^2=h_1^2+h_2^2. \end{align*} Thus \begin{align*} f(a+h)-f(a)-Df_a(h)=|h|^2. \end{align*} For $h\ne0$, the Euclidean norm satisfies $|h|>0$, and the error ratio becomes \begin{align*} \frac{|f(a+h)-f(a)-Df_a(h)|}{|h|}=\frac{||h|^2|}{|h|}. \end{align*} Since $|h|^2\ge0$, this is \begin{align*} \frac{|f(a+h)-f(a)-Df_a(h)|}{|h|}=\frac{|h|^2}{|h|}=|h|. \end{align*} As $h\to0$ in the Euclidean metric, $|h|\to0$, so \begin{align*} \frac{|f(a+h)-f(a)-Df_a(h)|}{|h|}\to0. \end{align*} The Euclidean metric supplies exactly the scale under which this quadratic error is smaller than the linear displacement. [/example] ### Coordinate Charts On manifolds, Euclidean distance is not usually available globally, but every coordinate chart lands in some $\mathbb{R}^n$. Local definitions of continuity, differentiability, and smoothness are expressed through Euclidean open sets and Euclidean limits inside chart images. This makes Euclidean balls the local measuring devices behind manifold topology. [example: Euclidean Balls in a Coordinate Chart] Let $M$ be a smooth manifold, let $(U,\varphi)$ be a chart with coordinates $(x_1,\dots,x_n)$, and fix $p\in U$. Write \begin{align*} \varphi(p)=(x_1(p),\dots,x_n(p)). \end{align*} Choose $r>0$ such that \begin{align*} B(\varphi(p),r)\subset \varphi(U). \end{align*} Since $d_2(\varphi(p),\varphi(p))=0<r$, we have $\varphi(p)\in B(\varphi(p),r)$, and therefore \begin{align*} p\in \varphi^{-1}(B(\varphi(p),r)). \end{align*} Set \begin{align*} V=\varphi^{-1}(B(\varphi(p),r)). \end{align*} Because $B(\varphi(p),r)\subset\varphi(U)$, every point of $V$ lies in $U$, so $V\subset U$. Moreover, for $q\in U$, \begin{align*} q\in V \iff \varphi(q)\in B(\varphi(p),r). \end{align*} By the Euclidean ball definition, this means \begin{align*} q\in V \iff d_2(\varphi(q),\varphi(p))<r. \end{align*} Writing $\varphi(q)=(x_1(q),\dots,x_n(q))$, the Euclidean metric formula gives \begin{align*} d_2(\varphi(q),\varphi(p))=\left(\sum_{k=1}^n |x_k(q)-x_k(p)|^2\right)^{1/2}. \end{align*} Thus $V$ consists exactly of the points whose coordinate vector lies within Euclidean radius $r$ of the coordinate vector of $p$. The restricted map \begin{align*} \varphi|_V:V\to B(\varphi(p),r) \end{align*} is bijective, because $\varphi$ is bijective from $U$ onto $\varphi(U)$ and $B(\varphi(p),r)\subset\varphi(U)$. Its inverse is the restriction of $\varphi^{-1}$ to $B(\varphi(p),r)$. Hence $V$ is a coordinate neighbourhood of $p$, with coordinates inherited from the original chart. Euclidean balls therefore become local neighbourhoods on $M$ by pulling them back through coordinate charts. [/example] This local role is why Euclidean metric geometry remains present even in differential geometry. Riemannian metrics generalize Euclidean inner products from $\mathbb{R}^n$ to tangent spaces, while charts still use Euclidean topology as their local target. ## Beyond and Connected Topics The Euclidean metric is the entry point to [metric spaces](/page/Metric%20Space), where the axioms of distance are studied without coordinates. Many Euclidean arguments survive in that setting, but theorems depending on linear structure, such as coordinatewise convergence or orthogonal symmetry, require additional hypotheses. It also leads naturally to [continuity](/page/Continuity) and multivariable differentiation. The $\varepsilon$-$\delta$ definition uses Euclidean balls in both domain and codomain, while differentiability measures the error term using Euclidean norms. In topology, the Euclidean metric generates the standard topology on $\mathbb{R}^n$. This connects directly to [open sets](/page/Open%20Set), compactness, connectedness, and the [product topology](/page/Product%20Topology) studied in [Cambridge IB Analysis and Topology](/page/Cambridge%20IB%20Analysis%20and%20Topology). In geometry, Euclidean distance is the flat model behind Riemannian geometry. A Riemannian metric assigns an inner product to each tangent space, and the resulting length and distance functions generalize the Euclidean formula. This direction continues in [Cambridge III Differential Geometry](/page/Cambridge%20III%20Differential%20Geometry) and [Cambridge III Riemannian Geometry](/page/Cambridge%20III%20Riemannian%20Geometry). In analysis and PDE, Euclidean balls define local estimates, supports, mollification regions, and Sobolev norms on domains $U\subset\mathbb{R}^n$. The metric is often invisible in notation, but it is present whenever a statement refers to local behaviour, bounded domains, or convergence of points. ## References Androma, [Cambridge IA Analysis Notes](/page/Cambridge%20IA%20Analysis%20Notes). Androma, [Metric Space](/page/Metric%20Space). Androma, [Continuity](/page/Continuity). Androma, [Open Set](/page/Open%20Set). Androma, [Cambridge IB Analysis and Topology](/page/Cambridge%20IB%20Analysis%20and%20Topology). Androma, [Cambridge III Differential Geometry](/page/Cambridge%20III%20Differential%20Geometry). Androma, [Cambridge III Riemannian Geometry](/page/Cambridge%20III%20Riemannian%20Geometry). Walter Rudin, *Principles of Mathematical Analysis* (1976). James R. Munkres, *Topology* (2000). Michael Spivak, *Calculus on Manifolds* (1965).