Line Integrals / Conservative Vector Fields

In summary, we are given two equations for F and asked to evaluate the integral of this function over a specific path from P to Q. In order to do so, we must specify that the path lies within the region where x, y, and z are all positive. This is necessary because the log function is only defined for positive values, and two of the elements in F involve the log function. When solving for the integral, we can use the given equations to find the values of f at the two points and then subtract them to get our final answer. As for the second question, it is likely related to the fact that the log function has a restricted domain and may not be defined for all paths.
  • #1
randomguy123
3
0

Homework Statement



[tex]F = < z^2/x, z^2/y, 2zlog(xy)>[/tex]
[tex]F = \nabla f[/tex], where [tex]f = z^2log(xy)[/tex]

Homework Equations



Evaluate [tex]\int F \cdot ds [/tex] for any path c from [tex] P = (1/2, 4, 2) [/tex] to [tex] Q = (2, 3, 3) [/tex] contained in the region [tex] x > 0, y > 0, z > 0 [/tex]

Why is it necessary to specify that the path lie in the region where [tex] x, y, z [/tex] are positive?

The Attempt at a Solution



I did [tex] f(2,3,3) - f(1/2,4,2) [/tex] to get [tex] 9*log(6) - 4*log(2) [/tex]

I don't really have an idea of how to answer the second question. Does it have to do with closed paths?
 
Last edited:
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  • #2
think about the domain of the log function.
 
  • #3
or the first 2 elements of F
 

Related to Line Integrals / Conservative Vector Fields

1. What is a line integral?

A line integral is a type of integral used in vector calculus to calculate the total value of a function along a given curve or path. It is used to measure the cumulative effect of a vector field along a specific direction.

2. What is a conservative vector field?

A conservative vector field is a type of vector field where the line integral is independent of the path taken. In other words, the value of the line integral remains the same regardless of the starting and ending points on the curve.

3. How do you calculate a line integral?

To calculate a line integral, you need to first parameterize the curve or path along which you want to integrate the function. Then, you need to multiply the function by the derivative of the parameterization and integrate it with respect to the parameter. The resulting value is the line integral.

4. What is the significance of conservative vector fields?

Conservative vector fields have many practical applications in physics and engineering. They are used to calculate work done by a force, potential energy, and fluid flow in a closed loop. They also have important implications in the study of conservative forces and conservative systems in physics.

5. How do you determine if a vector field is conservative?

A vector field is conservative if its line integral is independent of the path taken. Mathematically, this can be determined by checking if the curl of the vector field is equal to zero. If the curl is zero, the vector field is conservative. Additionally, if the vector field is the gradient of a scalar function, it is also conservative.

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