Let $\mathbf{x}$ be a closed path. Suppose that $\mathbf{F}$ represents the velocity vector field of a fluid. Consider the amount of fluid moved tangentially along a small segment of the path $\mathbf{x}$ during a brief time interval $\Delta \tau$. Since $\mathbf{F}\cdot \mathbf{T}$ gives the tangential component of $\mathbf{F}$, we have that:
$\qquad(1)\quad\qquad$Amount of fluid moved $\approx (\mathbf{F}(\mathbf{x}(t))\Delta \tau \cdot \mathbf{T}(t))\Delta s$
where $t$ is the parameter variable of the path $\mathbf{x}$ and $\Delta s$ is the length of the segment of the closed path $\mathbf{x}$. If we divide the term in $(1)$ by $\Delta \tau$, then the average rate of transport along the segment during the time interval $\Delta \tau$ is $(\mathbf{F}(\mathbf{x}(t)) \cdot \mathbf{T}(t))\Delta s$. If we now partition the closed path $\mathbf{x}$ into finitely many such small segments and sum the contributions of the form $(\mathbf{F}(\mathbf{x}(t)) \cdot \mathbf{T}(t))\Delta s$ for each segment, then let all the lengths $\Delta s$ tend to zero, we find that the average rate of fluid moved, denoted $\Delta L/\Delta \tau$, is given approximately by ("approximately" is what my book says):$$\frac{\Delta L}{\Delta\tau}\approx\int_{\mathbf{x}}(\mathbf{F}\cdot \mathbf{T})\,ds$$ Finally, if we let $\Delta \tau \to0,$ we may define the instantaneous rate of fluid moved, $dL/d\tau$, to be:$$\frac{dL}{d\tau}=\int_{\mathbf{x}}(\mathbf{F}\cdot \mathbf{T})\,ds=\int_{\mathbf{x}}\mathbf{F}\cdot d\mathbf{s}$$ Now let me ask you some questions: why does the circulation measure the instantaneous rate of fluid? What does this mean? Why can't you put the equal sign before let $\Delta\tau \to 0$?
My book says: "Assume that $\mathbf{F}$ does not vary with time". But if it doesn't vary, shouldn't the latter expression mean "$dL/d\tau$ is constant"? And then the amount of fluid moved should be directly proportional to time, shouldn't it?
The vector line integral introduction explains how the line integral ∫CF⋅ds of a vector field F over an oriented curve C “adds up” the component of the vector field that is tangent to the curve. In this sense, the line integral measures how much the vector field is aligned with the curve. If the curve C is a closed curve, then the line integral indicates how much the vector field tends to circulate around the curve C.