Let ω be a k-form and let η be a 1-form. Find d(don η-ω^ dn)
1. Let ω be a k-form in Rn , Π 〉 k. If k is odd, show that ωΛω 0 4. L et ω be a k-form and let ) be a /-form. Fin d d(da) Λ η_ωΛ đ7) .
1. Let ω be a k-form in Rn , Π 〉 k. If k is odd, show that ωΛω 0 4. L et ω be a k-form and let ) be a /-form. Fin d d(da) Λ η_ωΛ đ7) .
prove
Σ() (")- ("). η +η - η m-k k=0
. Show that, for every η Σ1: Τη Σ2, η(η + 1)(2n + 1) 6 k=1
5. Let S = {vi, u2, , v) be a set of k vectors in Rn with k > n. Show that S cannot be a basis for Rn.
Problem statement: Prove the following: Theorem: Let n, r, s be positive integers, and let v1, . . . , vr E Rn and wi, . . . , w, є Rn. If wi є span {v1, . . . , vr} for each i = 1, . . . , s, then spanfVi, . .., v-) -spanfvi, . .., Vr, W,...,w,) Suggestiorn: To see how the proof should go, first try the case s - 1, r 2..]
Problem...
Problem 3: Let xi be given n mutually orthogonal vectors in Rn, and 20 є Rn be also given. Find: (a) the distance di from Zo to Hi-{x E Rn : XTXǐ (b) the distance sk from ro to n1Hi, 1 <k< n (c) the distance mk from a'0 to ngk+1H,, 1-K n (d) calculate sk + mk 0)
Problem 3: Let xi be given n mutually orthogonal vectors in Rn, and 20 є Rn be also given. Find: (a)...
(1) Let w1, be a k-form and w2 be an l- form, both defined in an open subset UC R3. Let d : /\k (U)-ל ЛК +1 (U) be the exterior derivative of differential forms. (a) Show that d is a linear transformation of vector spaces. (b) Show that (c) Show that (d) Show that d(w) -d(d(w)) 0 for every k-form w, i.e. the map is the zero map
(1) Let w1, be a k-form and w2 be an l-...
1. Let {rn;n > 1} be a sequence of real numbers such that rn → x, where r is real. For each n let yn = (1/n) E*j. Show that yn + x. HINT: (xj – a) Let e >0 and use the definition of convergence. Split the summation into two parts and show that each is < e for all sufficiently large n.
8. Let w be the differential form yzdydz + zxdzdx + xydrdy. (i) Show that w is closed. (ii) Is w exact? If it is, find η such that dr-w. If not, explain why not.
8. Let w be the differential form yzdydz + zxdzdx + xydrdy. (i) Show that w is closed. (ii) Is w exact? If it is, find η such that dr-w. If not, explain why not.