What is the line integral in higher dimensions?

Question

Given a function $\ f:\mathbb{R}^2\to\mathbb{R}$ and some curve $\ \gamma:[a,b]\to\mathbb{R}^2$ it is my understanding that the integral of $\ f$ over $\ \gamma$ is the area of the region "between" $\ \gamma$ and $f(\gamma)\ $. Starting with the idea of trying to calculate the area of this region, you do the usual partitioning of $\ [a,b]$ and draw line segments between sample points on $\ \gamma$. Using the mean value theorem and taking the limit you get the formula for this area to be $$\int_a^bf(\gamma(t))\left|\gamma^\prime(t)\right|dt$$

For arbitrary $n$ and $\ f :\mathbb{R}^n\to\mathbb{R}$ and $\ \gamma:[a,b]\to\mathbb{R}^n$, one just defines the integral of $\ f$ over $\ \gamma$ to be this above formula. But what is this representing and how do we know it is giving something useful? Is there some physical intuition behind the line integral if $n>2$?

Answer 1

Yes. In $\mathbb{R}^2$, $\int_C f(x,y)\,ds = \int_a^b f({\bf r}(t))|{\bf r}'(t)|\,dt$ (where $C$ is parameterized by ${\bf r}(t)$, $a \leq t \leq b$) gives essentially the "area under the curve".

More precisely, we can visualize this as: $C$ traces out some curve in the $xy$-plane and directly above that $C$ cuts out a corresponding curve on the surface $z=f(x,y)$. The "sheet" below the curve on $z=f(x,y)$ and above $C$ has area $\int_C f(x,y)\,ds$.

The exact same thing is true in higher dimensions. $C$ traces out a curve in $\mathbb{R}^n$ and "above" it we get a corresponding curve on the hypersurface $z=f(x_1,x_2,\dots,x_n)$. $\int_C f(x_1,\dots,x_n)\,ds$ is then the area under this curve on the hypersurface and above $C$ in the $x_1\cdots x_n$-hyperplane.

So $f(x_1,\dots,x_n)$ gives a "height" and $ds$ gives a "width".

The only real different is that we cannot visualize this anymore.

In the end, this is probably not the best way to think about these integrals. Instead it tends to be more helpful thinking about $\int_C f({\bf x})\,ds$ as a sort of weighted sum along the curve $C$. Specifically, think of $f({\bf x})$ as being the weight/importance/density per unit length at the point ${\bf x}$. Then $f({\bf x})\,ds$ is the weight/importance/mass for along a "tiny chunk" of $C$. So $\int_C f({\bf x})\,ds$ is the total weight/importance/mass along the whole curve $C$.

Answer 2

"Area under the curve" is only one interpretation of an integral, and not necessarily the best. Computing work done to an object in physics is a good example to keep in mind. Another way line integrals arise is when computing the amount that a function $f:\mathbb R^n \to \mathbb R$ changes as its input moves along a curve $C$ connecting $a$ to $b$. Chop up the curve into tiny pieces connecting $x_i$ to $x_{i+1}$. Then \begin{align*} f(b) - f(a) &= \sum_i f(x_{i+1}) - f(x_i) \\ &\approx \sum_i \langle \nabla f(x_i), \Delta x_i \rangle \\ &\approx \int_C \nabla f \cdot dx. \end{align*} (Here $\Delta x_i = x_{i+1} - x_i$.)