How Do You Calculate a 3D Line Integral Along Multiple Paths?

In summary, the user is asking for help with calculating a line integral using three different routes and is specifically struggling with determining the limit in the 3D case. They mention using an example from a book for a 2D case and request assistance with the first part while they will complete the other two parts on their own.
  • #1
extreme2000
8
0
Dear Users
Please help me in starting this problem I have tried my best but all in vain
Calculate line integral v=X^2{x(Cap)}+2yz{y(Cap)}+y^2{z(Cap)} from origion to point
(1,1,1) by three different routes
(a) (0,0,0)→(1,0,0)→(1,1,0)→(1,1,1)
Now there are three parts in this problem.I want help in what will I consider the limit in 3D case because in example of the line integral of the book (Introducation to Electrodynamics by Griffiths) there is 2D example solved so I want a little help about this part I will do the other 2 parts myself but need help in first part
Thanks
 
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  • #2
Does v look like -
[tex]v\,=\,x^2\,\hat{x}+2yz\,\hat{y}+y^2\,\hat{z}[/tex]?
 

What is a line integral?

A line integral is a mathematical concept used in vector calculus to calculate the total value of a function along a given curve or path. It is a way to measure the change of a function in multiple dimensions and is often used in physics and engineering.

What is the formula for calculating a line integral?

The formula for calculating a line integral is ∫C F(x,y) ds, where C represents the curve or path, F(x,y) is the function being integrated, and ds represents the differential element of arc length along the curve.

How do you choose the curve for a line integral?

The curve for a line integral is typically given in the problem you are trying to solve. It could be a straight line, a curve defined by a function, or a closed loop. It is important to pay attention to the direction of the curve, as this will affect the value of the line integral.

What is the difference between a line integral and a double integral?

A line integral is calculated along a curve or path, while a double integral is calculated over a two-dimensional region. Line integrals are used to measure change along a curve, while double integrals are used for finding areas and volumes.

What are some real-world applications of line integrals?

Line integrals have many applications in various fields, such as calculating work done by a force, finding the mass and center of mass of an object, and determining the circulation of a vector field. They are also used in electromagnetism, fluid dynamics, and computer graphics.

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