I have a scalar function that takes $n$ arguments, $f(x_1, x_2,x_n) = f(\mathbf{x})$, and two vectors also with $n$ elements, $\mathbf{z} = (z_1, z_2\cdots,, z_n)$, and $\Delta\mathbf{z} = (\Delta z_1, \Delta z_2,\cdots ,\Delta z_n)$. Basically, I want to write the integral:
$$\int_{\mathbf{z+\Delta z}}^{\mathbf{z}}f(\mathbf{x})d\mathbf{x} = \int_{z_1+ \Delta z_1}^{z_1}\int_{z_2+ \Delta z_2}^{z_2}f(x_1, x_2,\cdots,x_n)\, dx_1 \, dx_2\,…dx_n$$
but for an arbitrary number of elements/arguments. I believe this is a standard multiple integral, though I've never seen one written using vectors for the limits and differentials; surface and volume integrals are usually expressed by writing a single integral sign for each dimension. Note: $\mathbf{z}$ and $\Delta \mathbf{z}$ are always greater than or equal to zero in my application.
Is this a vaid way of writing this integral, or am I going about this wrong?
Best Answer
I would write something like $\int_{\prod_{i=1}^{n}[z_i+\Delta z_i,z_i]}f(x) dx $ or define $Q := \prod_{i=1}^{n}[z_i+\Delta z_i,z_i]$ and write $\int_{Q}f(x) dx $.