Multivariable Advanced Calculus

11.10. EXERCISES 323the right thing for n vectors. In doing this, you might want to show that a vectorwhich is perpendicular to the span of v 1 , · · · , v n−1 is⎛⎞u 1 u 2 · · · u nv 11 v 12 · · · v 1ndet ⎜⎝..⎟. ⎠v n−1,1 v n−1,2 · · · v n−1,nwhere {u 1 , · · · , u n } is an orthonormal basis for span (v 1 , · · · , v n ) and v ij is thej th component of v i with respect to this orthonormal basis. Then argue thatif you replace the top line with v n1 , · · · , v nn , the absolute value of the resultingdeterminant is the appropriate definition of the volume of the parallelepiped. Nextnote you could get this number by taking the determinant of the transpose of theabove matrix times that matrix and then take a square root. After this, identifythis product with a Grammian matrix and then the desired result follows.5. Why is the definition of area on a manifold given above reasonable and what isits geometric meaning? Each functionu i → R −1 (u 1 , · · · , u n )yields a curve which lies in Ω. Thus R −1,u iis a vector tangent to this curve andR −1,u idu i is an “infinitesimal” vector tangent to the curve. Now use the previousproblem to see that when you find the area of a set on Ω, you are essentiallysumming the volumes of infinitesimal parallelepipeds which are “tangent” to Ω.6. Let Ω be a bounded open set in R n with P C 1 boundary or more generally one forwhich the divergence theorem holds. Let u, v ∈ C 2 ( Ω ) . Then∫Ω∫(v∆u − u∆v) dx =∂Ω(v ∂u∂n − u ∂v )dσ n−1∂nHere∂u∂n ≡ ∇u · nwhere n is the unit outer normal described above. Establish this formula which isknown as Green’s identity. Hint: You might establish the following easy identity.∇ · (v∇u) − v∆u = ∇v · ∇u.Recall ∆u ≡ ∑ nk=1 u x k x kand ∇u = (u x1 , · · · , u xn ) while∇ · F ≡ f 1x1 + · · · + f nxn ≡ div (F)