Line integration homework question

In summary, the problem involves two expressions (df=ydx-xdy and dg=ydx+xdy), one of which is an exact differential and the other is not. The task is to evaluate the line integral for each expression over three different paths: a straight line from (1,1) to (2,2), a combination of two straight lines from (1,1) to (2,1) and from (2,1) to (2,2), and a closed loop consisting of straight lines from (1,1) to (2,1) to (2,2) to (1,1). The student is unsure of how to approach the problem and is seeking help.
  • #1
amenforramen
1
0

Homework Statement


Consider two expressions (df=ydx-xdy) and (dg=ydx+xdy). One is an exact differential, the other is not. Evaluate the line integral for each over the following paths:

a) in a straight line form (1,1) to (2,2)
b) a straight line from (1,1) to (2,1), then a straight line from (2,1) to (2,2)
c) A closed loops consisting of straight lines from (1,1) to (2,1) to (2,2) to (1,1)


Homework Equations





The Attempt at a Solution


a)
Well I know dy/dx = 1, so y=x for this first equation... so I think from (1,1) to (2,2) ydx-xyd = 0? ... euh
 
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  • #2
h I'm not sure how to find the line integral for this, and I'm also not sure if my first equation is correct.... any help would be much appreciated :)
 

1. What is line integration and why is it important in scientific research?

Line integration is a mathematical technique used in physics, engineering, and other scientific fields to calculate the total value of a function along a given path or line. It is important because it allows scientists to analyze and understand the behavior of physical systems and make predictions about their future behavior.

2. How is line integration different from regular integration?

Regular integration involves calculating the area under a curve, while line integration involves calculating the line integral of a vector field along a given path. Line integration also takes into account the direction and orientation of the path, whereas regular integration does not.

3. What are the different types of line integrals?

The two main types of line integrals are the path integral and the contour integral. The path integral calculates the line integral along a given path, while the contour integral calculates the line integral over a closed curve.

4. How is line integration used in real-world applications?

Line integration is used in a variety of real-world applications, such as calculating the work done by a force on an object, finding the electric potential of a charged particle, and determining the flow of a fluid through a surface. It is also used in the fields of computer graphics, computer vision, and image processing.

5. What are some common challenges when solving line integration problems?

Some common challenges when solving line integration problems include determining the correct path or curve to integrate along, setting up the correct integrand, and dealing with complex or multi-dimensional vector fields. It is also important to pay attention to the orientation of the path and ensure that it is consistent with the direction of the vector field.

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