Path-independence implies no circulation

If the vector field F is a path-independent vector field, then the path integral

$$\int_{{C}}^{}$$

does not depend on the actual path C; the path integral depends only on the beginning point (call it p) and end point (call it q) of the path C.

Suppose, for example, we have two paths C₁ and C₂ connecting point p with point q.

Then we know that

$$\int_{{C_1}}^{} \int_{{C_2}}^{}$$ (1)

because F is path-independent.

Now, what would happen is we turned the path C₂ around, so that it starts at point q and ends at point p? (We will denote this path as C₂^-.) At any point, the tangent vector of C₂^- will be the opposite of the tangent vector of C₂ because the path is going in the opposite direction. Consequently

$$\int_{{C_2^-}}^{} \int_{{C_2}}^{}$$ (2)

If we define the path C to be the path C₁ followed by the path C₂^-, the path C will start at the point p, go to the point q (via C₁) and then return to the point p (via C₂). In other words, the path C will be a closed path.

Then path integral over C is simply the path integral over C₁ plus the path integral over C₂^-. By combining equation (1) with equation (2), we see these last two integrals are opposites

$$\int_{{C_2^-}}^{} \int_{{C_1}}^{}$$

We conclude that the path integral over C is zero:

$$\int_{{C}}^{} \int_{{C_1}}^{} \int_{{C_2^-}}^{}$$

If C is a closed path, we call the integral

$$\int_{{C}}^{}$$

the circulation of F around C. If F represents fluid flow, this integral indicates the tendancy for the fluid to circulate around the curve C. We could use the above argument to show that F is path-independent if and only if the circulation around any closed curve is zero.

We can use this result as a test for path-dependence. If we can find a single path C where

$$\int_{{C}}^{} \ne$$

then we know that F is path-dependent. For the example vector field F(x, y) = (y, - x) shown at the end of the previous reading, one can see the nonzero circulation around any circular path centered at the origin. This observation is enough to conclude that F is path-dependent.

The key point to remember is that path-independence means there is no circulation around any curve.