Does the integral depend on the path?

and that the value of the line integral depends only on the two endpoints, not on the path. The line integral is said to be independent and F is a conservative field.

What does independent of path mean?

the property of a function for which the line integral has the same value along all curves between two specified points.

What function is path independent?

A vector field is path-independent if and only if the circulation around every closed curve in its domain is 0. 0 . If a vector field F is path-independent, then there exists a function f such that ∇f=F. That is, F is a conservative or gradient vector field.

How do I find my path in Independence?

Independence of Path Theorem Let F(r) be continuous on an open connected set D. Then ∫F(r)·dr is independent of any path, C, in D iff F(r)=∇f(r) for some f(r) (scalar function), i.e. if F(r) is a conservative vector field on D. Let F(r) be continuous on an open connected set D. The following statements are equivalent.

When an integral is path independent?

An integral is path independent if it only depends on the starting and finishing points. Consequently, on any curve C={r(t)|t∈[a,b]}, by the fundamental theorem of calculus ∫CFdr=∫C∇fdr=f(r(b))−f(r(a)), in other words the integral only depends on r(b) and r(a): it is path independent.

Is a line integral path independent?

Path independence In other words, the integral of F over C depends solely on the values of G at the points r(b) and r(a), and is thus independent of the path between them. For this reason, a line integral of a conservative vector field is called path independent.

Why is integral path independent?

An integral is path independent if it only depends on the starting and finishing points. in other words the integral only depends on r(b) and r(a): it is path independent.

How that the line integral is independent of path and evaluate the integral?

The theorem tells us that in order to evaluate this integral all we need are the initial and final points of the curve. This in turn tells us that the line integral must be independent of path. If ∫C→F⋅d→r=0 ∫ C F → ⋅ d r → = 0 for every closed path C then ∫C→F⋅d→r ∫ C F → ⋅ d r → is independent of path.

What is a path independent line integral?

About Transcript. Showing that if a vector field is the gradient of a scalar field, then its line integral is path independent.

How do you know if an integral is independent of path?

Showing that if a vector field is the gradient of a scalar field, then its line integral is path independent.

How do you know if a line integral is independent of path?

The theorem tells us that in order to evaluate this integral all we need are the initial and final points of the curve. This in turn tells us that the line integral must be independent of path. If →F is a conservative vector field then ∫C→F⋅d→r ∫ C F → ⋅ d r → is independent of path.

What is meant by line integral and path?

A line integral (sometimes called a path integral) is the integral of some function along a curve. One can integrate a scalar-valued function along a curve, obtaining for example, the mass of a wire from its density. One can also integrate a certain type of vector-valued functions along a curve.

How is the line integral independent of path?

The theorem tells us that in order to evaluate this integral all we need are the initial and final points of the curve. This in turn tells us that the line integral must be independent of path. If →F F → is a conservative vector field then ∫ C →F ⋅ d→r ∫ C F → ⋅ d r → is independent of path. This fact is also easy enough to prove.

What is the theorem of the line integral?

The theorem tells us that in order to evaluate this integral all we need are the initial and final points of the curve. This in turn tells us that the line integral must be independent of path. If →F F → is a conservative vector field then ∫ C →F ⋅ d→r ∫ C F → ⋅ d r → is independent of path.

When is the line integral of a vector field the same?

No matter which path you follow between two points in a conservative vector field, whether it’s a direct, straight line, or a curvy, winding path, or any other path, the value of the line integral will be the same if the endpoints are the same.

Which is independent of the path C C?

∫ C →F ⋅ d→r ∫ C F → ⋅ d r → is independent of path if ∫ C1 →F ⋅ d →r = ∫ C2 →F ⋅ d→r ∫ C 1 F → ⋅ d r → = ∫ C 2 F → ⋅ d r → for any two paths C1 C 1 and C2 C 2 in D D with the same initial and final points. A path C C is called closed if its initial and final points are the same point. For example, a circle is a closed path.