infinitesimal

Examples of infinitesimal in the following topics:

• Arc Length and Speed

  • A curve may be thought of as an infinite number of infinitesimal straight line segments, each pointing in a slightly different direction to make up the curve.
  • In order to calculate the arc length, we use integration because it is an efficient way to add up a series of infinitesimal lengths.
  • The arc length is calculated by laying out an infinite number of infinitesimal right triangles along the curve.
  • Each of these triangles has a width $dx$ and a height $dy$, standing for an infinitesimal increase in $x$ and $y$.

• Arc Length and Surface Area

  • Infinitesimal calculus provides us general formulas for the arc length of a curve and the surface area of a solid.
  • The advent of infinitesimal calculus led to a general formula, which we will learn in this atom.
  • Consider an infinitesimal part of the curve $ds$ (or consider this as a limit in which the change in $s$ approaches $ds$).

• Curl and Divergence

  • The curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field.
  • If $\hat{\mathbf{n}}$ is any unit vector, the projection of the curl of $\mathbf{F}$ onto $\hat{\mathbf{n}}$ is defined to be the limiting value of a closed line integral in a plane orthogonal to $\hat{\mathbf{n}}$ as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.
  • More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
  • It is a local measure of its "outgoingness"—the extent to which there is more exiting an infinitesimal region of space than entering it.

• Differentials

  • In physical applications, the variables $dx$ and $dy$ are often constrained to be very small ("infinitesimal").

• Cylindrical Shells

  • The idea is that a "representative rectangle" (used in the most basic forms of integration, such as $\int x \,dx$) can be rotated about the axis of revolution, thus generating a hollow cylinder with infinitesimal volume.

• Linear Approximation

  • If one were to take an infinitesimally small step size for $a$, the linear approximation would exactly match the function.

• Area and Distances

  • The key idea is the transition from adding a finite number of differences of approximation points multiplied by their respective function values to using an infinite number of fine, or infinitesimal, steps.
  • Consider an infinitesimal part of the curve $ds$ on the curve (or consider this as a limit in which the change in $s$ approaches $ds$).

• The Fundamental Theorem of Calculus

  • Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities and introduced the notation used today.

• Volumes of Revolution

  • The volume of each infinitesimal disc is therefore:

• The Definite Integral

  • The key idea is the transition from adding a finite number of differences of approximation points multiplied by their respective function values to using an infinite number of fine, or infinitesimal, steps.