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I am wondering if there is a mean value theorem for line integral. For example, let $f(x):\mathbb{R}^n\rightarrow \mathbb{R}$ be a continuous (not necessarily monotonic) function defined on smooth curve $C$, do we have the following theorem:

$F(x)=\int_Cf(x)dx=f(x_0)\cdot L_C$

where $x_0$ is some point on the curve $C$ (i.e., $x_0\in C$ ) and $L_C$ is the length of the curve $C$?

In addition, if $f(x)=\nabla F(x)$, do we have $F(a)-F(b)=||f(x_0)||\cdot L_C$, where $||\cdot||$ is some norm (e.g., $\ell_2$ norm)?

Thanks.

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The answer is yes for continuous functions $f$ on continuous curves $C$. The reason for this is that the line integral (with respect to arc length) satisfies the inequality

$$ \min(f) \cdot L_C \leq \int_C f(x_1, \ldots x_n) \,ds \leq \max(f) \cdot L_C $$

which can be rearranged as

$$ \min(f) \leq \cfrac{1}{L_C} \int_C f(x_1, \ldots x_n) \,ds \leq \max(f) $$

We know that the minimum and maximum of the continuous function $f$ on $C$ exist because $C$ is compact, being the continuous image of $[0, 1]$. But $C$ is also connected for the same reason, so the range of $f$ must be an interval. The only option is that the range of $f$ must be the interval $[\min(f), \max(f)]$.

This is important because this number $\cfrac{1}{L_C} \int_C f(x_1, \ldots x_n)$ was just shown to be in that interval, and therefore is in the range of $f$. So, pick a value $\vec{c} \in C$ with $f(\vec{c}) = \cfrac{1}{L_C} \int_C f(x_1, \ldots x_n)$ and the theorem is proved.

EDIT: The added part of your question about the integral of a gradient vector field $f(x) = \nabla F(x)$ will have a negative answer. For example, a line integral of a gradient across a loop (initial point = endpoint) will give you zero, but the gradient vector field you are integrating might never be zero.

Example: Integrate the gradient of $F(x, y) = x + y$ along the counterclockwise oriented unit circle.

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  • $\begingroup$ Thanks for the answer, @Ben Passer! How about if $f(x)=\nabla F(x)$? Does the statement still hold: $F(a)-F(b)=||\nabla f(x_0)||\cdot L_C$? $\endgroup$ – user3138073 Dec 29 '14 at 4:15
  • $\begingroup$ I think there may have been some confusion. The line integral above is the integral of a real-valued function with respect to arc length, not the integral of a vector field. You should not expect to see mean value theorems for integrals of vector fields. Especially take a look at the right hand side of your edit. That quantity is always a positive number! See the edit to my answer. $\endgroup$ – Ben Passer Dec 29 '14 at 5:35
  • $\begingroup$ you are right. Thanks! $\endgroup$ – user3138073 Dec 29 '14 at 13:36

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