How do i maximize the line integral?

• okkvlt
In summary, the conversation discusses the determination of the path taken from point A to point B in a nonconservative vector field in order to maximize the line integral. However, it is mentioned that this may be an undefined problem without constraints on the coordinates and length of the curve. Additionally, the possibility of circling against the flow to maximize the line integral is brought up, which requires knowledge of the Euler-Lagrange equation as it is a "calculus of variations" problem.
okkvlt
suppose i have a nonconservative vector field.
and there is a path going from point A to point B.
How do i determine the path taken from A to B such that the line integral is maximized?
edit: actually after thinkin about it, this might be an undefined problem unless there is some constraint on the x,y,z coordinates.

Last edited:
okkvlt said:
actually after thinkin about it, this might be an undefined problem unless there is some constraint on the x,y,z coordinates.

And on the length of the curve. Suppose a vector field circles the destination counterclockwise. What's to stop you from circling against the flow, round and round...

That is a "calculus of variations" problem. Do you know about the Euler-Lagrange equation?

1. How do I calculate the line integral?

The line integral can be calculated using the formula ∫ab F(x,y) ds, where F(x,y) is the function being integrated, a and b represent the starting and ending points on the curve, and ds is the length element.

2. What is the purpose of maximizing the line integral?

Maximizing the line integral allows you to find the maximum value of a given function along a curve or path. This can be useful in various applications, such as finding the maximum work done by a force along a certain path.

3. What is the difference between a line integral and a regular integral?

A line integral is calculated along a curve or path, while a regular integral is calculated over a specific interval. Line integrals also take into account the direction of the curve, while regular integrals do not.

4. How do I choose the correct path for maximizing the line integral?

The path chosen for maximizing the line integral depends on the given function and the starting and ending points. It is important to choose a path that is continuous and differentiable in the given interval.

5. Can the line integral be negative?

Yes, the line integral can be negative if the function being integrated is negative along the chosen path. This indicates that the work done by the force along the path is negative.

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