potential energy

(noun)

the energy possessed by an object because of its position (in a gravitational or electric field), or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)

Examples of potential energy in the following topics:

  • The Chain Rule

    • $U$ could be electric potential energy at a location $(x,y)$.
    • What we want to calculate is the rate of change of the potential energy $U$ as a function of time $t$.

  • Conservative Vector Fields

    • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
    • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
    • When the above equation holds, $\varphi$ is called a scalar potential for $\mathbf{v}$.

  • 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:

  • Fundamental Theorem for Line Integrals

    • By placing $\varphi$ as potential, $\nabla$ is a conservative field.
    • Electric field is a vector field which can be represented as a gradient of a scalar field, called electric potential.

  • Essential Functions for Mathematical Modeling

    • Another example is a model of a particle in a potential-field.
    • The potential field is given by a function $V:R^3 \rightarrow R$ and the trajectory is a solution of a differential equation.

  • Vector Fields

    • When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and, under this interpretation, conservation of energy is exhibited as a special case of the fundamental theorem of calculus.

  • Work

    • Work done by the restoring force leads to increase in the kinetic energy of the object.

  • Stokes' Theorem

    • The scalar field $\varphi$ in the case of electromagnetism is called electric potential.