dimension

Examples of dimension in the following topics:

• Dimensional Analysis

  • This is often used to represent the dimension of individual basic quantity.
  • An example of the use of basic dimensions is speed, which has a dimension of 1 in length and -1 in time; $\displaystyle \frac{[L]}{[T]} = [LT^{-1}]$.
  • The dimension of any physical quantity is the combination of the basic physical dimensions that compose it.
  • The dimensions of derived quantities may include few or all dimensions in individual basic quantities.
  • where n represents the amount per u dimensions.

• Spaces Associated with a linear system Ax = y

• Area Expansion

  • Objects expand in all dimensions.
  • We learned about the linear expansion (in one dimension) in the previous Atom.
  • Objects expand in all dimensions, and we can extend the thermal expansion for 1D to two (or three) dimensions.
  • The area thermal expansion coefficient relates the change in a material's area dimensions to a change in temperature.
  • The change in the linear dimension can be estimated as: $\frac{\Delta A}{A} = \alpha_A \Delta T$.

• Length

  • Length is one of the basic dimensions used to measure an object.
  • In geometric measurements, length is the longest dimension of an object.
  • In other contexts "length" is the measured dimension of an object.
  • Length is a measure of one dimension, whereas area is a measure of two dimensions (length squared) and volume is a measure of three dimensions (length cubed).

• Inelastic Collisions in Multiple Dimensions

  • At this point we will expand our discussion of inelastic collisions in one dimension to inelastic collisions in multiple dimensions.
  • Relate inelastic collision multiple dimension equations to the one dimension collisions you learned earlier

• Elastic Collisions in Multiple Dimensions

  • If an elastic collision occurs in two dimensions, the colliding masses can travel side to side after the collision (not just along the same line as in a one dimensional collision).
  • The general approach to finding the defining equations for an n-dimensional elastic collision problem is to apply conservation of momentum in each of the n- dimensions.
  • In this illustration, we see the initial and final configurations of two masses that undergo an elastic collision in two dimensions.
  • A brief introduction to problem solving of collisions in two dimensions using the law of conservation of momentum.

• Introduction to Waves and Modes in One and Two Spatial Dimensions

• Linear Expansion

  • To a first approximation, the change in length measurements of an object (linear dimension as opposed to, for example, volumetric dimension) due to thermal expansion is related to temperature change by a linear expansion coefficient.
  • where L is a particular length measurement and dL/dT is the rate of change of that linear dimension per unit change in temperature.
  • From the definition of the expansion coefficient, the change in the linear dimension $\Delta L$ over a temperature range $\Delta T$ can be estimated to be:

• Four-Dimensional Space-Time

  • (See for an example. ) Therefore both observers live in a four-dimensional world with three space dimensions and one time dimension.
  • You should not find it odd to work with four dimensions; any time you have to meet your friend somewhere you have to tell him four variables: where (three spatial coordinates) and when (one time coordinate).
  • In other words, we have always lived in four dimensions, but so far you have probably thought of space and time as completely separate.

• Linear Dependence and Independence

  • of the same dimension.
  • Linear independence is also central to the notion of how big a vector space is--its dimension.
  • So the dimension of a space is the number of linearly independent vectors required to represent an arbitrary element.