Recently I was reading about triple integrals and I came across the statement -
"We saw that a double integral could be thought of as the volume under a two-dimensional surface. It turns out that the triple integral simply generalizes this idea: it can be thought of as representing the hypervolume under a three-dimensional hypersurface w=f(x,y,z) whose graph lies in $R^4$"
My doubts are:
- We imagine or assume Volume in $R^3$ as space enclosed by a curve. But what can be imagined for a hypervolume.
- We can calculate Area in $R^2$ as well as $R^3$. So in $R^4$, using same analogy can we calculate Volume of curves in $R^4$.
- Are volume and hypervolume different quantities in $R^4$?
I'm in a confused state with all these questions. Pl. Help me.
Thanks for everyone in advance
A square has 1-dimensional edges, which have lengths.
A cube has 1d edges (which have lengths) and 2d faces (which have area).
A hypercube is a 4 dimensional object. It is bounded by several cubes (similar to how a cube is bounded by many faces). These bounding cubes have volume. The hypercube has a hypervolume of 1*1*1*1 (it is 1 unit long in each of the 4 directions).
This corresponds to the case of integrating w=f(x,y,z)=1 over $0\leq x,y,z\leq1$.