# Understanding Independence of Path in Calculus 3

• livenn
In summary, the concept of independence of path states that for a continuous and conservative function F in an open region R, the line integral over any piecewise smooth curve C from one fixed point in R to another fixed point in R is the same. This means that the value of the integral only depends on the end points and not the path taken. This can be visualized by imagining a particle moving between two potential surfaces, where the work done on it only depends on the end points and not the path taken.
livenn
Hi,
I'm having a bit of difficulty wrapping my mind around the concept of independence of path. My textbook says:

If F is continuous and conservative in an open region R, the value of int(F.dr) over the curve C is the same for every piecewise smooth curve C from one fixed point in R to another fixed point in R. This result is described by saying that the line intint(F.dr) over the curve C is independent of path in the region R.

I get the continuous and conservative on the open region part...

But I'm failing to comprehend how this means that

the value of int(F.dr) over the curve C is the same for every piecewise smooth curve C from one fixed point in R to another fixed point in R

This really makes no sense to me, and I don't have a visual image of this to consult, could anyone enlighten me, or does anyone know of any good images or applets that explain this clearly? Unfortunately I haven't found any on google. Thanks in advance.

Edit: Just as a point of reference, this is for calculus 3.

OK I've done a crappy paint drawing, I hope it helps:

What it is saying is, for a conservative field, the integral over C1 = the integral over C2 = int over C3 = int over C4, because the value over the integral only depends on the end points R1 and R2, ie it is I(R2)-I(R1), where I'=F.

Think of a particle moving between two potential surfaces. If the force is conservative, and if the particle moves along an equipotential surface, no work is done on it. If it moves between potential surfaces, the work done on it only depends on the amount it has moved along the gradient of the potential. So you might as well only consider where it starts and where it ends, to find the distance along the potential gradient it has moved.

## 1. What is independence of path in Calculus 3?

Independence of path in Calculus 3 refers to the concept that the path of integration does not affect the value of the line integral. This means that the line integral will have the same value regardless of the path taken between the two endpoints.

## 2. Why is understanding independence of path important in Calculus 3?

Understanding independence of path is important in Calculus 3 because it allows us to simplify complex line integrals by choosing a path that is easier to integrate. It also helps in finding the exact value of a line integral without having to evaluate it for every possible path.

## 3. How is independence of path different from linearity in Calculus 3?

Independence of path and linearity are closely related concepts in Calculus 3, but they are not the same. Linearity refers to the property that allows us to split a line integral into smaller, simpler integrals. Independence of path, on the other hand, refers to the fact that the path of integration does not affect the value of the line integral.

## 4. Can you give an example of independence of path in action?

One example of independence of path is the line integral of a conservative vector field. It has been proven that the value of a line integral of a conservative vector field is independent of the path taken between the two endpoints. This allows us to use simpler paths to evaluate the integral and still get the same result.

## 5. Are there any exceptions to independence of path in Calculus 3?

Yes, there are some cases where independence of path does not hold true. For example, if the vector field is not continuous or if the path of integration crosses a singularity or discontinuity, then the value of the line integral may be affected by the path. In such cases, it is important to carefully choose the path of integration to get an accurate result.

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