# Definition:Euclidean Metric

## Definition

Let $M_{1'} = \struct {A_{1'}, d_{1'} }$ and $M_{2'} = \struct {A_{2'}, d_{2'} }$ be metric spaces.

Let $A_{1'} \times A_{2'}$ be the cartesian product of $A_{1'}$ and $A_{2'}$.

The **Euclidean metric** on $A_{1'} \times A_{2'}$ is defined as:

- $\map {d_2} {x, y} := \paren {\paren {\map {d_{1'} } {x_1, y_1} }^2 + \paren {\map {d_{2'} } {x_2, y_2} }^2}^{1/2}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_{1'} \times A_{2'}$.

### General Definition

The **Euclidean metric** on $\ds \AA = \prod_{i \mathop = 1}^n A_{i'}$ is defined as:

- $\ds \map {d_2} {x, y} := \paren {\sum_{i \mathop = 1}^n \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2}$

where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$.

### Riemannian Manifold

Let $x \in \R^n$ be a point.

Let $\tuple {x_1, \ldots, x_n}$ be the standard coordinates.

Let $T_x \R^n$ be the tangent space of $\R^n$ at $x$.

Let $T_x \R^n$ be identified with $\R^n$:

- $T_x \R^n \cong \R^n$

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Let $v, w \in T_x \R^n$ be vectors such that:

- $\ds v = \sum_{i \mathop = 1}^n v^i \valueat {\partial_i} x$

- $\ds w = \sum_{i \mathop = 1}^n w^i \valueat {\partial_i} x$

Let $g$ be a Riemannian metric such that:

- $\ds g_x = \innerprod v w_x = \sum_{i \mathop = 1}^n v^i w^i$

Then $g$ is called the **Euclidean metric**.

## Special Cases

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### Real Number Plane

The **Euclidean metric** on $\R^2$ is defined as:

- $\ds \map {d_2} {x, y} := \sqrt {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.

### Rational Number Plane

The **Euclidean metric** on $\Q^2$ is defined as:

- $\ds \map {d_2} {x, y} := \sqrt {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \Q^2$.

### Complex Plane

The **Euclidean metric** on $\C$ is defined as:

- $\d \forall z_1, z_2 \in \C: \map d {z_1, z_2} := \size {z_1 - z_2}$

where $\size {z_1 - z_2}$ denotes the modulus of $z_1 - z_2$.

## Also known as

The **Euclidean metric** is also known as the **Euclidean distance**.

Some sources refer to it as the **Cartesian distance**, for RenĂ© Descartes.

## Also see

- Results about
**the Euclidean metric**can be found here.

## Source of Name

This entry was named for Euclid.

## Historical Note

Euclid himself did not in fact conceive of the Euclidean metric and its associated Euclidean space, Euclidean topology and Euclidean norm.

They bear that name because the geometric space which it gives rise to is **Euclidean** in the sense that it is consistent with Euclid's fifth postulate.

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Pythagoras' Theorem - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.7$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Euclidean distance**or**Cartesian distance** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Euclidean metric** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Euclidean distance (Cartesian distance)**