Line integral curiosity/confusion

In summary, A line integral can be represented graphically by thinking of it in terms of work done under a force. However, the fact that the line integral of a function over a curve is 0 does not necessarily mean that the function must be identically 0 on that curve. This is because the function may be positive on one part of the curve and negative on another, or it may be non-zero elsewhere. Therefore, it is important to work with the definitions instead of trying to visualize the concept.
  • #1
optinyx
2
0
Heya!

I was hoping someone could clear this up for me: how would a line integral be represted graphically? I've always liked calculus because it's easy to visualize (almost all the problems have graphs associated with them) - but I don't quite get how to visualize a line integral. Or is it a concept that can't be easily visualized?

any help would, well.. help. :redface:

I've also been pondering something my teacher brought up last week:
Let's say the line integral of f(x,y,z) over a curve C is equal to zero, then would it have to be true that f(x,y,z) = 0 on C?

Thank you! I eagerly await a response. :biggrin:
 
Physics news on Phys.org
  • #2
its easier to visualize it with physics... think of it in terms of Work done under a force. On the other q, you mean for ALL C's?, because if C is closed, then is not necesarily true, think of it in terms of the work done :)
 
  • #3
No, the fact that the line integral of f over a curve is 0 does NOT mean that the function must be identically 0 for two reasons:

First it is possible that f is positive on part of the curve and negative on another part so that the two parts cancel.
Also, knowing the integral on a particular curve only gives you information about f ON THAT CURVE. Even if f were 0 on the curve itself, it might be non-zero elsewhere.
 
  • #4
okay, I think I understand now. thinking of it in terms of work done makes it easier, like in physics.. when you push a block ten feet in one direction and then ten feet back to where you started, they say the total work done was zero because the total distance was zero. Or am I completely off my rocker? (it's been a while since general physics).

If I understand though, then that's what hallsofivy was saying right? - about it being positive on one part and negative on the other? I think I get it.

Could anyone give me an example function, f(x,y,z), and a curve where this would be the case?

Thank you for all the help.
 
  • #5
optinyx said:
okay, I think I understand now. thinking of it in terms of work done makes it easier, like in physics.. when you push a block ten feet in one direction and then ten feet back to where you started, they say the total work done was zero because the total distance was zero. Or am I completely off my rocker? (it's been a while since general physics).

displacement*
 
Last edited:
  • #6
try to stop thinking visually, it doesn't help in the real world, or the world of mathematics, just work with the definitions.
 

1. 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 takes into account both the function value and the direction of movement along the curve.

2. How is a line integral different from a regular integral?

A regular integral calculates the area under a curve, while a line integral calculates the value of a function along a curve. In other words, a regular integral is a two-dimensional concept, while a line integral is a one-dimensional concept.

3. What is the significance of a line integral in real-world applications?

Line integrals have a wide range of applications in various fields such as physics, engineering, and economics. They are used to calculate work done by a force, electric and magnetic fields, fluid flow, and many other physical quantities.

4. How do I calculate a line integral?

To calculate a line integral, you need to first parameterize the curve or path along which the integral is to be evaluated. Then, you need to plug in the values of the curve parameters into the given function and integrate over the specified interval. You can use various integration techniques such as substitution and partial fractions to solve the integral.

5. What are some common misconceptions about line integrals?

One common misconception is that line integrals can only be calculated in Cartesian coordinates. In fact, they can also be evaluated in other coordinate systems such as polar, cylindrical, and spherical coordinates. Another misconception is that line integrals only apply to straight lines, but they can also be applied to any smooth curve or path.

Similar threads

Replies
10
Views
3K
Replies
8
Views
1K
Replies
24
Views
2K
Replies
3
Views
217
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
955
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
12
Views
936
Back
Top