Stokes' theorem

(noun)

a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.

Related Terms

Examples of Stokes' theorem in the following topics:

  • Changing Magnetic Flux Produces an Electric Field

    • Using the Stokes' theorem in vector calculus, the left hand side is$\oint_C \vec E \cdot d\vec s = \int_S (\nabla \times \vec E) \cdot d\vec A$.

  • Poincaré Sphere

    • This result shows that the Stoke's parameters live on a sphere of radius $r\leq s_0$ where the extent of polarization $\Pi=r/s_0$.
    • which relates Stoke's parameters to the orientation and shape of the polarization ellipse.
    • An interesting and useful relationship is that the Stokes parameters are additive for waves whose phases are not correlated.
    • Let's take two waves of frequencies $\omega_a$ and $\omega_b$ and calculate the value of the first Stokes parameter as an example.
    • When we measure the Stokes parameters in practice we measure for example

  • Stokes Parameters

    • The four Stokes parameters satisfy the following relationship for a truly monochromatic wave

  • Kinetic Energy and Work-Energy Theorem

    • The work-energy theorem states that the work done by all forces acting on a particle equals the change in the particle's kinetic energy.
    • The principle of work and kinetic energy (also known as the work-energy theorem) states that the work done by the sum of all forces acting on a particle equals the change in the kinetic energy of the particle.
    • This relationship is generalized in the work-energy theorem.

  • Gravitational Attraction of Spherical Bodies: A Uniform Sphere

    • The Shell Theorem states that a spherically symmetric object affects other objects as if all of its mass were concentrated at its center.
    • That is, a mass $m$ within a spherically symmetric shell of mass $M$, will feel no net force (Statement 2 of Shell Theorem).
    • We can use the results and corollaries of the Shell Theorem to analyze this case.
    • This diagram outlines the geometry considered when proving The Shell Theorem.
    • (Note: The proof of the theorem is not presented here.

  • Convergence Theorems

  • A.2 Parseval's Theorem

  • Some Basic Theorems for the Fourier Transform

    • And finally, we have the convolution theorem.
    • The convolution theorem is one of the most important in time series analysis.
    • The convolution theorem is worth proving.

  • Gauss's Law

    • Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field.
    • The law can be expressed mathematically using vector calculus in integral form and differential form, both are equivalent since they are related by the divergence theorem, also called Gauss's theorem.

  • Spaces Associated with a linear system Ax = y