Stokes' Theorem
Stokes' Theorem - Related Videos
01:06
Divergence and Stokes' Theorems
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2025
The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write numerous physical laws...
01:24
Stokes’ Theorem and Its Applications
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2026
Stokes’ Theorem provides a fundamental connection between the circulation of a vector field along a closed boundary and the cumulative rotational behavior across the surface it encloses. For a smooth three-dimensional surface with an oriented boundary curve, this theorem offers a unified way to relate motion along the edge to local rotational effects distributed over the surface.Mathematical FormulationThe theorem states that the circulation of a vector field along a closed curve is equal to...
01:20
Stokes' Law
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2025
Viscous forces, like friction, are intermolecular forces that resist the relative motion of molecules over each other. When a solid body moves through a liquid, viscous forces drag it in the opposite direction. The force's magnitude depends on the solid's shape and size, as well as its speed and the liquid's coefficient of viscosity, density and temperature. The expression for the force on a solid spherical object in a fluid is called Stokes' law. Stokes' law is valid only for low Reynolds...
03:00
Coherent Anti-Stokes Raman Spectroscopy to Visualize Myelinated Neurons in Mouse Brain Tissue
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2025
Source: McCullagh, E. A., et al. Coherent Anti-Stokes Raman Spectroscopy (CARS) Application for Imaging Myelination in Brain Slices. J. Vis. Exp. (2022)The video demonstrates fluorescence and Coherent Anti-Stokes Raman Spectroscopy (CARS) imaging to visualize neuronal architecture in rodent brain tissue. Nissl staining labels cell bodies by binding to ribosomal RNA, while CARS imaging targets lipid-rich myelin through laser-induced signals. Overlaying the fluorescence and CARS images provides a...
12:21
Stimulated Stokes and Antistokes Raman Scattering in Microspherical Whispering Gallery Mode Resonators
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Cited by 1 •
2016
Efficient generation of nonlinear phenomena related to third order optical non-linear susceptibility Χ(3) interactions in triply resonant silica microspheres is presented in this paper. The interactions here reported are: Stimulated Raman Scattering (SRS), and four wave mixing processes comprising Stimulated Anti-stokes Raman Scattering (SARS).
FAQs
The orientation links the direction of traversal along C to the chosen normal direction of S. Apply the right-hand rule: if the fingers follow the boundary direction, the thumb indicates the corresponding surface orientation. Consistent direction choices are essential because the theorem connects the circulation around the boundary with the curl evaluated using that surface orientation.
Stokes' Theorem separates a field's behavior into two related views. The circulation records the cumulative effect along the boundary, while the curl describes the field's local contribution throughout the surface. Integrating that local quantity over S connects it to the boundary result, making the theorem useful when local field behavior is easier to analyze than the enclosing curve.
The theorem provides two mathematically connected routes for the same circulation result. If the boundary calculation is difficult, one can work with the curl across the enclosed surface; if the surface calculation is less convenient, the boundary perspective may be preferable. The useful choice depends on which representation makes the field behavior easier to handle.
A practical setup begins by identifying the vector field, the surface S, and its closed boundary C. Next, choose compatible orientations using the right-hand rule, then write the circulation as a line integral or the curl contribution as a surface integral. Comparing these equivalent forms helps organize the calculation and check the intended boundary-surface relationship.
Applications center on situations where circulation matters. In magnetic-field analysis, the theorem relates field behavior across a surface to effects traced around its boundary. In fluid-flow analysis, it provides the same local-to-boundary viewpoint for circulation. These uses show why the result is valuable beyond formal calculus: it connects field descriptions with effects measured or interpreted along boundaries.
The surface S supplies the region over which the curl is accumulated and must correspond to the closed boundary C. Its orientation also determines how the boundary is traversed through the right-hand rule. Selecting and describing the surface is therefore part of setting up the theorem, because it organizes the relationship between the surface calculation and the boundary circulation.