How to Visualize Minimizing Definite Integrals and Understand Stationary Points?

In summary, Calculus of variations is a branch of mathematics that deals with finding the minimum or maximum of definite integrals. It involves working with expressions as functions and applying calculus methods to find stationary points. This concept is similar to minimization in multivariable calculus, but with an infinite number of dimensions.
  • #1
richardlhp
13
0
Calculus of variations (HELP!)

Hi all! Just a question...

How should I visualise geometrically the minimising of definite integrals, and what is the significance of finding stationary points of definite integrals? (Can someone provide me with an intuitive explanation?)

Thanks so much!
 
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  • #2


Assuming you mean the definite integral has a parameter, work with the expression as a function of the parameter and then apply elementary calculus methods. Being a definite integral is not particularly relevant.
 
  • #3


He probably means variations of functions and not parameters.

Intuitively, it's similar to minimization in multivariable calculus, but with an infinite number of dimensions.

OK, the mathematicians should be sufficiently stunned that I have a few seconds to duck for cover.
 
  • #4


Thanks so much!
 

FAQ: How to Visualize Minimizing Definite Integrals and Understand Stationary Points?

What is the purpose of calculus of variations?

The purpose of calculus of variations is to find the function or curve that minimizes or maximizes a given functional. This can be applied to real-world problems in physics, engineering, economics, and other fields.

What is a functional in calculus of variations?

A functional is a mathematical expression that takes in a function as an input and outputs a number. It represents a quantity that is dependent on the shape of the function, rather than just its value at a point.

What is the Euler-Lagrange equation in calculus of variations?

The Euler-Lagrange equation is a necessary condition for finding the optimal function in calculus of variations. It states that the derivative of the functional with respect to the function must be equal to zero at the optimal function.

What is the difference between constrained and unconstrained problems in calculus of variations?

In constrained problems, there are certain limitations or conditions that the optimal function must satisfy, while in unconstrained problems, there are no such limitations. This can affect the approach and solution methods used in calculus of variations.

What are some real-world applications of calculus of variations?

Calculus of variations has been used to solve problems in a variety of fields, including finding the shortest path between two points, optimizing the shape of a bridge or building, and determining the path of a projectile to hit a target. It is also used in optimal control theory, which has applications in engineering, economics, and biology.

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