Examples of euclidean space in the following topics:

 The most familiar examples are those that arise as the boundaries of solid objects in ordinary threedimensional Euclidean space $R^3$— for example, the surface of a ball.
 On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in threedimensional Euclidean space without introducing singularities or selfintersections.
 Historically, surfaces were initially defined as subspaces of Euclidean spaces.
 Such a definition considered the surface as part of a larger (Euclidean) space, and as such was termed extrinsic.

 A parametric surface is a surface in the Euclidean space $R^3$ which is defined by a parametric equation.
 A parametric surface is a surface in the Euclidean space $R^3$ which is defined by a parametric equation with two parameters: $\vec r: \Bbb{R}^2 \rightarrow \Bbb{R}^3$.

 The shape of an object is a description of space that the object takes up; the shape can change if the object is deformed.
 The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as color, content, and material composition.
 In geometry, two subsets of a Euclidean space have the same shape if one can be transformed to the other by a combination of translations, rotations (together also called rigid transformations), and uniform scalings.
 Having the same shape is an equivalence relation, and accordingly a precise mathematical definition of the notion of shape can be given as being an equivalence class of subsets of a Euclidean space having the same shape.
 In particular, the shape does not depend on the size and placement in space of the object.

 A vector field is an assignment of a vector to each point in a subset of Euclidean space.
 In vector calculus, a vector field is an assignment of a vector to each point in a subset of Euclidean space.
 Vector fields are often used to model the speed and direction of a moving fluid throughout space, for example, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point.
 Vector fields can be thought to represent the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (the rate of change of volume of a flow) and curl (the rotation of a flow).

 The gravitational potential associated with a mass distribution given by a mass measure $dm$ on threedimensional Euclidean space $R^3$ is:
 If there is a continuous function $\rho(x)$ representing the density of the distribution at $x$, so that $dm(x) = \rho (x)d^3x$, where $d^3x$ is the Euclidean volume element, then the gravitational potential is:

 A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane, known as the axis .

 The graph of this function defines a surface in Euclidean space .

 A Euclidean vector is a geometric object that has magnitude (i.e. length) and direction.
 A Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction and can be added to other vectors according to vector algebra.
 A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point $A$ with a terminal point $B$, and denoted by $\vec{AB}$.
 For instance, in three dimensions, the points $A=(1,0,0)$ and $B=(0,1,0)$ in space determine the free vector $\vec{AB}$ pointing from the point $x=1$ on the $x$axis to the point $y=1$ on the $y$axis.
 A vector in the 3D Cartesian space, showing the position of a point $A$ represented by a black arrow.

 A norm is a function from the space of vectors onto the scalars, denoted by $\ \cdot \$ satisfying the following properties for any two vectors $v$ and $u$ and any scalar $\alpha$ :
 For $p=2$ this is just the ordinary Euclidean norm: $\\mathbf{x}\ _ 2 = \sqrt{\mathbf{x}^T \mathbf{x}}$ .

 This kind of a matrix is the starting point for almost all network analysis, and is called an "adjacency matrix" because it represents who is next to, or adjacent to whom in the "social space" mapped by the relations that we have measured.
 But social distance can be a funny (nonEuclidean) thing.