Question: If M is an dimensional manifold
If M is an -dimensional manifold (or manifold-with-boundary) in R n, with the usual orientation, show that ∫ fdx1 ^ . ^ dx n, as defined in this section, is the same as ∫ M f, as defined in Chapter 3.
Answer to relevant Questionsa. Show that Theorem 5-5 is false if M is not required to be compact. b. Show that Theorem 5-5 holds for noncom-pact M provided that w vanishes outside of a compact subset of M.If M is an -dimensional manifold in Rn, with the usual orientation, show that dV = dx1^ . . . ^ dxn, so that the volume of M, as defined in this section, is the volume as defined in Chapter 3. (Note that this depends on the ...Let f, g: [0, 1] → R3 be nonintersecting closed curves. Define the linking number l (f, g) of and g by (cf. Problem 4-34A formula for estimating the mean free path of a perfect gas is: ℓ = 1.26 μ/p√R|T) = 1.26 μ/p √ (RT) Where the latter form follows from the ideal-gas law, ρ = p/RT. What are the dimensions ...The volume flow Q over a dam is proportional to dam width B and also varies with gravity g and excess water height H upstream, as shown in Fig. P1.14. What is the only possible dimensionally homogeneous relation for this ...
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