Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

Castigliano's Theorem

Castigliano's Theorem

Castigliano's theorem analyzes displacements and rotations in elastic structures. It relates the derivative of elastic strain energy to the applied forces or moments, allowing for the calculation of deformations. The theorem states that the partial derivative of the total strain energy of a system with respect to a specific load results in the displacement at the point where the load is applied. This principle applies to both forces and moments.

Theorems of Pappus and Guldinus

Theorems of Pappus and Guldinus

The two theorems developed by Pappus and Guldinus are widely used in mathematics, engineering, and physics to find the surface area and volume of any body of revolution. This is done by revolving a plane curve around an axis that does not intersect the curve to find its surface area or revolving a plane area around a non-intersecting axis to calculate its volume.
For finding the surface area, consider a differential line element that generates a ring with surface area dA when revolved.

Theorem of Pappus

Theorem of Pappus

The Theorem of Pappus, also known as the Pappus–Guldinus Theorem, provides a geometric method for determining the volume and surface area of solids generated by the revolution of a plane region or a plane curve about an external axis. The theorem consists of two related statements. The first addresses the volume of solids formed by rotating plane areas, while the second addresses the surface area generated by rotating plane curves. Both results depend on the location of the centroid, which...

Theorems of Pappus and Guldinus: Problem Solving

Theorems of Pappus and Guldinus: Problem Solving

Pappus and Guldinus's theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For example, consider a cylindrical storage tank with a conical top. Finding the surface area or volume can be challenging for such complex shapes. These theorems are particularly useful in calculating the volume and surface area of such systems. Here, the cylindrical storage tank with a conical top can be broken down into two simple shapes: a cylinder...

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as: with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the Complete Factorization...

Castigliano's Theorem: Problem Solving

Castigliano's Theorem: Problem Solving

The deflection of a simply supported beam that carries a central point load can be analyzed using structural mechanics principles, particularly by applying Castigliano's theorem. This theorem relates the displacement at the load application point to the partial derivatives of the strain energy in the structure. The simply supported beam with a point load at its center has symmetric reaction forces at the supports, each bearing half of the load. The bending moment at any point along the beam is...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by

Same author

MARCHF6 orchestrates hepatic lipid homeostasis by targeting SREBP1 for ER-associated degradation.

Journal of hepatology·2026

Same author

A new paradigm: 46 structural cell types function as environment-supporting innate immune cells, with endothelial and vascular smooth muscle cells as key prototypes.

iNew medicine·2026

Same author

Existence of Finite Time Blow-Up in Keller-Segel System.

Annals of PDE·2026

Same author

Androgen excess uncouples circulating and hepatic lipid homeostasis in females on a high-calorie diet.

Journal of lipid research·2026

Same author

Long-term trends in intensive care unit hospitalizations in Poland: a study based on the National Hospital Registry, 2012-2021.

Archives of medical science : AMS·2026

Same author

Self-sustaining humidity gradient in nanocellulose moist-electric transparent generator via hydrophilic-hydrophobic Janus interfaces.

Carbohydrate polymers·2026

Search research articles

Related Experiment Video

Updated: May 23, 2026

Scalable Quantum Integrated Circuits on Superconducting Two-Dimensional Electron Gas Platform

Scalable Quantum Integrated Circuits on Superconducting Two-Dimensional Electron Gas Platform

Published on: August 2, 2019

On De Giorgi's conjecture and beyond.

Manuel Del Pino¹, Michal Kowalczyk, Juncheng Wei

¹Departamento de Ingeniería Matemática and Center for Mathematical Modeling, Universidad de Chile, Casilla 170/3, Santiago, Chile.

Proceedings of the National Academy of Sciences of the United States of America

|April 14, 2012

Summary

This study finds a counterexample to a major conjecture in phase transition modeling. The Allen-Cahn equation solutions challenge existing theories on minimal surfaces and entire solutions.

More Related Videos

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

Published on: August 12, 2013

Related Experiment Videos

Last Updated: May 23, 2026

Scalable Quantum Integrated Circuits on Superconducting Two-Dimensional Electron Gas Platform

Scalable Quantum Integrated Circuits on Superconducting Two-Dimensional Electron Gas Platform

Published on: August 2, 2019

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

Published on: August 12, 2013

Area of Science:

Nonlinear Partial Differential Equations
Geometric Analysis
Phase Transitions

Background:

The Allen-Cahn equation is a key model for phase transitions.
Its entire solutions are linked to minimal surface theory.
A celebrated question by Ennio de Giorgi concerned these solutions.

Purpose of the Study:

To investigate the existence of entire solutions to the Allen-Cahn equation.
To address De Giorgi's conjecture regarding minimal entire graphs.
To explore the relationship between Allen-Cahn solutions and minimal surfaces.

Main Methods:

Exploiting the connection between the Allen-Cahn equation and minimal surface theory.
Analyzing solutions in dimensions N ≥ 9.
Constructing a specific solution with ∂(x(N))u > 0.

Main Results:

A counterexample to De Giorgi's conjecture is found.
The solution's level sets approximate a nonplanar, minimal, entire graph.
This result is valid for dimensions N ≥ 9.

Conclusions:

The existence of certain entire solutions to the Allen-Cahn equation is confirmed.
De Giorgi's conjecture is disproven.
The findings suggest further research into finite Morse index solutions and classification of all entire solutions.