Recent content by ttsp

What are the A_n? Are they cubes used to approximate A? How do you know that we can actually approximate A arbitrarily close?

That's a good hint. I know how to prove it when A is a cube, and I can see intuitively why that would imply the same when A is an arbitrary area. However, how do I show that mathematically? Suppose that I was able to show the statement for any cube R. How do I generalize to any arbitrary region A?

I don't know the Stokes' theorem. Is it possible to prove it without the Stokes' theorem? Can I do it straight from the definition? Does the volume element matter? It can be any dx^i . A can be any arbitrary bounded region, not necessarily a cube.

Indeed I can only use the definition. I would first create a rectangle R such that A is included in R and \tilde{f} such that \tilde{f}(x) = 1 if x is in A and \tilde{f}(x) = 0 if x is in R\A And by definition, I have: \int_{A} 1 = \int_{R} \tilde{f} = sup \{L(\tilde{f},P) \} . How...

Homework Statement Show that \int_{A} 1 = \int_{T(A)} 1 given A is an arbitrary region in R^n (not necessarily a rectangle) and T is a translation in R^n. Homework Equations Normally we find Riemann integrals by creating a rectangle R that includes A and set the function to be zero when x...

Homework Statement Find a piecewise smooth parametric curve to the astroid. The astroid, given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$, is not smooth, as we see singular points at 0, pi/2, 3pi/2, and 2pi. However is there a piecewise smooth curve? Homework Equations $\phi(\theta) =...