Line integrals and paths with the same endpoints

In summary, the conversation discusses a proof involving open, path-connected, and simply connected subsets of Rn and smooth curves with specific start and end points. The goal is to show that the line integral of a 1-form over two different curves is equal. The conversation also mentions the definition of "one-form" and the concept of an "exact differential." It is mentioned that the statement may not be true without additional conditions.
  • #1
Aaronc
1
0

Homework Statement


Suppose that p and q are points in U, where U is an open, path-connected, simply connected subset of Rn and c1 and c2 are smooth curves in Rn with c1(0)=c2(0)=p, c1(1)=c2(1)=q. Let w be a 1-form on U. Prove that the line integral of w over c1 equals the line integral of w over c2.


Homework Equations


simply connected means every closed curve homotopes to a point



I don't know where to start :(
 
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  • #2
Aaronc said:

Homework Statement


Suppose that p and q are points in U, where U is an open, path-connected, simply connected subset of Rn and c1 and c2 are smooth curves in Rn with c1(0)=c2(0)=p, c1(1)=c2(1)=q. Let w be a 1-form on U. Prove that the line integral of w over c1 equals the line integral of w over c2.


Homework Equations


simply connected means every closed curve homotopes to a point



I don't know where to start :(

Reading your textbook or lecture notes would be a good place to begin.

RGV
 
  • #3
You might start with the definition of "one-form". And do you know what an "exact differential" is? It looks to me like the statement you are trying to prove is NOT true unless there are some other conditions.
 

FAQ: Line integrals and paths with the same endpoints

What is a line integral?

A line integral is a type of integral used in multivariable calculus that involves integrating a function along a curve or path. It is used to calculate the amount of a given quantity that flows or accumulates along a given path.

What is the difference between a path and a curve?

A path is a continuous route or journey that connects two points, while a curve is a continuous and smooth line that may or may not be a path. In mathematics, a curve is a general term that can refer to any type of continuous line, while a path is a more specific term that refers to a specific route between two points.

What is the significance of having the same endpoints in a line integral?

The endpoints of a line integral are important because they define the start and end points of the path along which the integral is being calculated. Having the same endpoints means that the path is closed, and the integral can be calculated as a loop rather than a one-way journey.

How do you calculate a line integral with the same endpoints?

To calculate a line integral with the same endpoints, you first need to parameterize the path using a variable such as t. Then, you integrate the function you want to evaluate along the path with respect to t, and evaluate the integral at the upper and lower limits of t to get the final result.

What are some real-world applications of line integrals with the same endpoints?

Line integrals with the same endpoints have many real-world applications, including calculating work done by a force along a specific path, calculating fluid flow through a curve or pipe, and measuring the amount of heat transferred along a specific path. They are also used in physics, engineering, and other fields to analyze and solve problems involving quantities that change along a given path.

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