infinitesimal

Examples of infinitesimal in the following topics:

• Changes in the Entropy of Surroundings

  • A change is said to occur reversibly when it can be carried out in a series of infinitesimal steps.
  • The reversible expansion of a gas can be achieved by reducing the external pressure in a series of infinitesimal steps; reversing any step will restore the system and its surroundings to their previous state.
  • Similarly, heat can be transferred reversibly between two bodies by changing the temperature difference between them in infinitesimal steps, each of which can be undone by reversing the temperature difference.

• 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 $ds$).

• Flux

  • The flux is simply the rate that energy passes through an infinitesimal area (imagine a small window).

• 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.

• Induced EMF and Magnetic Flux

  • For a varying magnetic field, we first consider the magnetic flux $d\Phi _B$ through an infinitesimal area element dA, where we may consider the field to be constant:
  • A generic surface, A, can then be broken into infinitesimal elements and the total magnetic flux through the surface is then the surface integral

• Internal Energy

  • We can calculate a small change in internal energy of the system by considering the infinitesimal amount of heat δQ added to the system minus the infinitesimal amount of work δW done by the system:

• Difference Quotients

  • If |ΔP| is infinitesimal, an infinitely small amount usually expressed in standard analysis as a limit, then ΔF(P) is known as an infinitesimal difference, with specific denotations of dP and dF(P).

• Rotational Kinetic Energy: Work, Energy, and Power

  • Additional friction of the two global tidal waves creates energy in a physical manner, infinitesimally slowing down Earth's angular velocity.

• Differentials

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