What is the interpretation of the fundamental theorem of line integrals?
By James Williams
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The fundamental theorem of line integrals is:
$$\int_{a}^{b} \nabla F \cdot \vec{dr} = F(r(b)) - F(r(a))$$ for some curve traced by $r$.
What is the intuition for why this is true?
The proof is straightforward, but I cannot extend it.
2 Answers
$\begingroup$From a mathematical point of view, this theorem is just a generalization of $$ \int_a^b f(t) \;dt =\int_a^b F'(t) \;dt =F(b)-F(a), $$ where $F(t)$ is an antiderivative of $f(t)$. This essentially says that to compute an integral of a function that has an antiderivative, it is sufficient to compute the difference of the antiderivative between both endpoints of the interval. If there is more than one variable, the concept of gradient is in some sense equivalent to the concept of derivation, so the theorem becomes $$ \int_C \vec{f}(x,y) \cdot d\vec{r} =\int_C \nabla F(x,y)\cdot d\vec{r} =F(\vec{r}(b))-F(\vec{r}(a)). $$
From a more physical point of view, this theorem states that the integral does not depend on the curve $C$, but only on the endpoints of $C$. In other words, you can integrate on the curve that you want, you will get the same result, provided the starting and stopping point remain the same. This is very powerful and useful in physics, for example when you need to compute the energy (or work) done by a vector field (for example gravity) to move an object from one end point to another. If the field can be expressed as the gradient of a function (it is the case for gravity), then you don't need to know the trajectory of the object, you only need to know the endpoints.
This is how the common formula $$ W_{\vec{P}}=mgh $$ is proved. The work done by the weight field $\vec{P}$ only depends on constants $m$ and $g$, and on the difference of altitude of the object, i.e. on the initial and final position of the object. Whatever the object did between these positions is not important.
$\endgroup$ $\begingroup$An example that I keep in mind for this is that of a force field that gives the resistance to infinitesimal motion at each point, such as that of gravity. The line integral then gives you the work done in moving along a given path. When the force field has a conservation law (hence the term “conservative”), the integral will depend only on the end points of the path.
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