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 numerical factor in the definition of w ^ n.)
Answer to relevant QuestionsGeneralize Theorem 5-6 to the case of an oriented (n - 1) -dimensional manifold in Rn. The generalization is w Є A n-1(Mx) defined bya. Show that this length is the least upper bound of lengths of inscribed broken lines.For the triangular element in Fig P1.3, show that a tilted free liquid surface, in contact with an atmosphere at pressure pa, must undergo shear stress and hence begin to flow. Test, for dimensional homogeneity, the following formula for volume flow Q through a hole of diameter D in the side of a tank whose liquid surface is a distance h above the hole position: Q = 0.68D2 √gh Where g is ...(“C” means computer-oriented, although this one can be done analytically.) A baseball, with m = 145 g, is thrown directly upward from the initial position z = 0 and Vo = 45 m/s. The air drag on the ball is CV2, where C ...
Post your question