Euclidean

Examples of Euclidean 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.

• 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 $\vec{AB}$.

• 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.

• 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 .