Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Gradient and Del Operator01:14

Gradient and Del Operator

4.9K
In mathematics and physics, the gradient and del operator are fundamental concepts used to describe the behavior of functions and fields in space. The gradient is a mathematical operator that gives both the magnitude and direction of the maximum spatial rate of change. Consider a person standing on a mountain. The slope of the mountain at any given point is not defined unless it is quantified in a particular direction. For this reason, a "directional derivative" is defined, which is a vector...
4.9K
Second Derivatives and Laplace Operator01:22

Second Derivatives and Laplace Operator

2.8K
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
2.8K
Second Derivatives and the Shape of a Graph01:29

Second Derivatives and the Shape of a Graph

249
The second derivative of a function provides essential information about a graph's curvature and how it changes over an interval. It helps determine whether a function is concave upward or concave downward and identifies points where the curvature changes. These properties are fundamental in analyzing real-world scenarios, such as changes in road elevation, population growth, and economic trends.A function f(x) is considered concave upward on an interval if its graph lies above all its tangent...
249
State Function, Exact and Inexact Differentials01:27

State Function, Exact and Inexact Differentials

111
A state function is a thermodynamic property that depends solely on the current state of a system, irrespective of its history or how it arrived at that state. These functions are represented by capital letters, such as U, H, and S, which stand for internal energy, enthalpy, and entropy, respectively.For instance, the value of internal energy depends on the system's state variables and remains unaffected by the process path. This means that whether the system underwent a linear process or a...
111
Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

4.6K
The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
4.6K
First Derivatives and the Shape of a Graph01:22

First Derivatives and the Shape of a Graph

211
In calculus, the concept of the first derivative plays a crucial role in understanding the behavior of a function over its domain. The first derivative, denoted as f’(x), provides insight into how a function changes at any given point, much like a cyclist adjusting speed along a winding trail. By analyzing the first derivative, mathematicians can determine where a function is increasing, decreasing, or reaching critical points.The first derivative provides a precise method for classifying...
211

You might also read

Related Articles

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

Sort by
Same author

Aromaticity-Induced Spin State Switching and High-Spin States in Non-Alternant Polyradicals.

Journal of computational chemistry·2026
Same author

π-π Stacking Determines the Selectivity of Unnatural DNA Base Pairs Even without Polymerase.

ACS physical chemistry Au·2026
Same author

High-Spin Porphyrin Polyradicals.

ACS omega·2026
Same author

Evaluating Protein Liquid Supplementation for Enhanced Protein Intake and Adherence at Short-Term After Metabolic and Bariatric Surgery: A Pilot Randomized Controlled Trial.

Obesity surgery·2025
Same author

Controlling molecular machines <i>via</i> optimally oriented external electric fields.

Chemical science·2025
Same author

Leap from Diradicals to Tetraradicals by Topological Control of π-Conjugation.

The Journal of organic chemistry·2024

Related Experiment Video

Updated: Mar 29, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180&#176; Curved Artery Test Section
11:00

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

12.0K

The Variational Structure of Gradient Extremals.

Josep Maria Bofill1, Wolfgang Quapp2, Marc Caballero1

  • 1Institut de Química Teòrica i Computacional, Universitat de Barcelona (IQTCUB) , C/Martí i Franquès, 1, 08028 Barcelona, Catalunya, Spain.

Journal of Chemical Theory and Computation
|November 24, 2015
PubMed
Summary
This summary is machine-generated.

Gradient extremals represent chemical reaction paths and possess variational properties. These curves can be minimal under specific conditions, relating turning points to potential energy surface features.

More Related Videos

Computational Modeling of Retinal Neurons for Visual Prosthesis Research - Fundamental Approaches
10:50

Computational Modeling of Retinal Neurons for Visual Prosthesis Research - Fundamental Approaches

Published on: June 21, 2022

2.3K

Related Experiment Videos

Last Updated: Mar 29, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180&#176; Curved Artery Test Section
11:00

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

12.0K
Computational Modeling of Retinal Neurons for Visual Prosthesis Research - Fundamental Approaches
10:50

Computational Modeling of Retinal Neurons for Visual Prosthesis Research - Fundamental Approaches

Published on: June 21, 2022

2.3K

Area of Science:

  • Chemical Physics
  • Computational Chemistry
  • Theoretical Chemistry

Background:

  • Reaction paths are crucial for understanding chemical dynamics.
  • Gradient extremals offer a potential representation of these paths.
  • Their variational nature and relation to potential energy surface (PES) features require detailed investigation.

Purpose of the Study:

  • To establish gradient extremals as valid representations of reaction paths.
  • To determine the conditions under which these curves exhibit minimal character.
  • To explore the relationship between turning points on these curves and critical points on the PES, such as valley-ridge inflection points.

Main Methods:

  • Mathematical analysis to prove the variational nature of gradient extremals.
  • Derivation of conditions for gradient extremals to represent minimal curves.
  • Comparative analysis of turning points on gradient extremals with specific PES features.

Main Results:

  • Gradient extremals are demonstrated to possess a variational character.
  • Conditions are identified for these curves to function as minimal curves.
  • Relationships are elucidated between turning points of gradient extremals and critical points on the potential energy surface, including valley-ridge inflection points.

Conclusions:

  • Gradient extremals provide a robust mathematical framework for describing reaction paths.
  • The variational and minimal properties of these curves offer deeper insights into reaction dynamics.
  • Understanding the connection between gradient extremals and PES critical points enhances the interpretation of chemical reaction mechanisms.