I am expecting that this is a very simple question but I can't seem to find a clear answer.
Suppose that we have a function $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$. In one dimension ($n=1$), I can bound the integral of $f$ as follows: $$ \int_a^b f(x)dx \leq |b-a| \sup_x f(x) $$ When $n>1$, can I do something similar for the contour integral $$ \oint_\mathcal{C} f(x) \cdot dx $$ where $\mathcal{C}$ is a smooth curve from $a\in\mathbb{R}^n$ to $b\in\mathbb{R}^n$? Intuitively, this would look like $$ \oint_\mathcal{C} f(x) \cdot dx \leq ||b-a|| \sup_{x\in\mathcal{C}} ||f(x)|| $$ where $||\cdot||$ is some norm. If this works, can I use any norm (I guess not...)? I'm particularly interested in the case in which $f$ is a conservative vector field. In that case, the integral over a closed curve ($a=b$) would yield 0 so that using $||b-a||$ in the inequality would make sense. Many thanks!