euclidean space

Examples of euclidean space in the following topics:

• Surfaces in Space

  • The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional 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 three-dimensional Euclidean space without introducing singularities or self-intersections.
  • 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.

• Parametric Surfaces and Surface Integrals

  • 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: $\vec r: \Bbb{R}^2 \rightarrow \Bbb{R}^3$.

• Vector Fields

  • 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).

• Applications of Multiple Integrals

  • The gravitational potential associated with a mass distribution given by a mass measure $dm$ on three-dimensional 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:

• Area of a Surface of Revolution

  • 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 .

• Partial Derivatives

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

• Vectors in Three Dimensions

  • 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.

• Three-Dimensional Coordinate Systems

  • The three-dimensional coordinate system expresses a point in space with three parameters, often length, width and depth ($x$, $y$, and $z$).
  • A three dimensional space has three geometric parameters: $x$, $y$, and $z$.
  • Also known as analytical geometry, this system is used to describe every point in three dimensional space in three parameters, each perpendicular to the other two at the origin.
  • This is a three dimensional space represented by a Cartesian coordinate system.

• Cylinders and Quadric Surfaces

  • A quadric surface is any $D$-dimensional hypersurface in $(D+1)$-dimensional space defined as the locus of zeros of a quadratic polynomial.
  • A quadric, or quadric surface, is any $D$-dimensional hypersurface in $(D+1)$-dimensional space defined as the locus of zeros of a quadratic polynomial.

• Volumes

  • Volume is the quantity of three-dimensional space enclosed by some closed boundary—for example, the space that a substance or shape occupies or contains.
  • One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in three-dimensional space.