Math 322 Assignment 6

Math 322 Assignment 6 March 26, 2018

Submit problems 3 and 5 on Tuesday, 27 March.

1. LetA⊂R² be the rectangle [−1/2,1/2]×[0,2π] andf :A→R³ be given by f(s, t) = tcos(s/2) + 1

u(s) +tsin(s/2)v where u(s) = cos(s),sin(s),0

and v= (0,0,1). Show thatM =f(A) is a manifold with boundary of dimension 2. What is∂M?

2. Consider the cylinderC={(x, y, z)∈R³|x²+y²= 1, −1/2≤z≤1/2}. Show thatC is a manifold with boundary of dimension 2. What is ∂M?

3. LetU ⊂Rⁿ be open andf :U →R^k be a submersion. Leta∈f(U) and M ={x∈U |f(x) =a}.

(a) Let p∈ M and (V, φ) a chart forM at p, such that φ(x) = pfor x∈ V. Show that f_∗◦φ_∗ : TxR^n−k→TaR^k is the zero map of vector spaces.

(b) Show thatT_pM = ker(f_∗:T_pRⁿ →T_aR^k).

4. Let f :R² →Rbe a C¹ function and Γ_f ⊂R³ be the graph of f as in problem 10 of assignment 9.

Then Γ_f is a manifold.

(a) Find the tangent space to Γ_f atp= (0,0, f(0,0)).

(b) Now letf(x, y) = cos(x²+y²)

1 +x²+y² graph this function and repeat part (a) for this function.

5. Consider the following primitive model for a solar system. A star is located at the originO= (0,0,0), and is stationary. A planetP is revolving around the star at distance 10r(at constant speed) along the x-y plane. A satelliteS ofP is revolving aroundP at a distancerfromP (with constant speed) and is always in the plane containing the z-axis andP. During one complete revolution ofP, S revolves 10 times aroundP. InitiallyP is at (10r,0,0) andS at (11r,0,0).

(a) Show that the trajectory ofS is a manifold of dimension 1 and give an atlas for it.

(b) Find the tangent spaces at the points where the three bodies are collinear.

Multilinear Algebra.

LetV be a vector space andT^k(V) denote the space ofk tensors overV. Λ^k(V)⊂T^k(V) is the subset of all alternatingk tensors.

1. Show thatT^k(V) forms a vector space and Λ^k(V) is a vector subspace.

2. Recall the tensor product; ifS∈T^k(V) andT ∈T^{`</sup>(V) thenS⊗T ∈T<sup>k+`}(V) is defined by (S⊗T)(v1, . . . , vk+`) =S(v1, . . . , vk)T(vk+1, . . . , vk+`).

Show that the tensor product satisfies the following properties:

(a) (associativity)S⊗(T⊗U) = (S⊗T)⊗U;

(b) (distributivity)S⊗(T₁+T₂) =S⊗T₁+S⊗T₂and (S₁+S₂)⊗T =S₁⊗T+S₂⊗T; (c) (homogeneity) (cS)⊗T =S⊗(cT) =c(S⊗T) for anyc∈R.

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(d) Give an example whereS⊗T 6=T ⊗S (so commutativity fails).

Let the dimension ofV benandv1, . . . , vn be a basis ofV. Ifφ1, . . . , φnis the dual basis forV^∗=T¹(V).

Then

φi₁⊗ · · · ⊗φi_k, 1≤i1, . . . , ik ≤n is a basis for T^k(V). HenceT^k(V) has dimensionn^k.

3. Recall that a tensor T ∈T²(V) is called an inner product ifT(v, w) =T(w, v) for all v, w ∈V and T(v, v)>0 for allv∈V andv6= 0. Show that there is a basisv1, . . . , vn ofV such that

T(vi, vj) =

1 i=j, 0 otherwise.

Such a basis is called an orthonormal basis forT.

(a) Let w₁, . . . , w_n be any basis ofV. Define w₁⁰ =w₁,w₂⁰ =w₂−T(w⁰₁, w2) T(w⁰₁, w⁰₁)w⁰₁, w⁰₃=w₃−T(w⁰₁, w3)

T(w⁰₁, w₁⁰)w₁⁰ −T(w₂⁰, w3)

T(w₂⁰, w₂⁰)w₂⁰ and so on. Then show thatT(w⁰_i, w⁰_j) = 0 ifi6=j.

(b) Letv_i= 1

pT(w⁰_i, w_i⁰)w⁰_ithen show that v₁, . . . , v_n is an orthonormal basis forT.

4. Recall that if we have a linear transformation f :V →W between vector spaces, then there is a map f^∗:T^k(W)→T^k(V).

(a) Show that f^∗ is a linear transformation.

(b) Show thatf^∗ω∈Λ^k(V) ifω ∈Λ^k(W), So it gives a linear transformationf^∗: Λ^k(W)→Λ^k(V).

(Here there is a convenient abuse of notation.)

(c) Iff is an isomorphism then show thatf^∗ is also an isomorphism.

(d) IfT is an inner product on V andv1, . . . , vn and orthonormal basis for T, then define the linear transformationg:V →Rⁿgiven byg(vi) =ei wheree1, . . . , en is the standard basis ofRⁿ. Show thatT =f^∗h, i, whereh, iis the standard inner product onRⁿ.

5. Recall that Alt :T^k(V)→T^k(V) defined as follows, ifT ∈T^k(V) the tensor Alt(T) does the following AltT(v₁, . . . , v_k) = 1

k!

X

σ∈Sk

sgn(σ)T(v_σ(1), . . . , v_σ(k)).

Show the following:

(a) Alt(T)∈Λ^k(V). (Hint. Composing a 2-cycle with a permutation changes the sign of the permu- tation.)

(b) Alt(ω) =ω ifω∈Λ^k(V). (Hint. Show that all the summands of Alt are the same.)

6. Ifω∈Λ^k(V) andη∈Λ<sup>`</sup>(V) then their wedge product is defined as follows ω∧η= (k+`)!

k!`! Alt(ω⊗η) Show that the wedge product satisfies the following properties:

(a) ω∧η = (−1)^k`η∧ω (graded commutativity);

(b) ω∧(η1+η2) =ω∧η1+ω∧η2 and (ω1+ω2)∧η =ω1∧η+ω2∧η (distributivity);

(c) (aω)∧η=ω∧(aη) =a(ω∧η) for anya∈R(homogeneity);

The wedge product also satisfies associativity but it is a bit harder to prove. Refer to Theorem 4-4 on page 80 of Spivak.

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Let the dimension ofV benandv1, . . . , vn be a basis ofV. Ifφ1, . . . , φn is the dual basis forV^∗ = Λ¹(V).

Then

φi₁∧ · · · ∧φi_k, 1≤i1< . . . < ik≤n is a basis for T^k(V). HenceT^k(V)has dimension ⁿ_k

. In particularΛⁿ(V)is 1-dimensional andΛ^k(V) is trivial if k > n.

7. Prove the following for the wedge product:

(a) IfφΛ¹(V) thenφ∧φ= 0. (Hint. Use problem 6 (a).) (b) Ifφ1, . . . , φk∈Λ¹(V) then

φ1∧. . .∧φk = X

σ∈S_k

sgn(σ)φ_σ(1)⊗ · · · ⊗φ_σ(n)

Recall that any non-zero element ω ofΛⁿ(V)gives an orientation of V. Ifv1, . . . , vn is a (ordered) basis of V then it is positively oriented if ω(v1, . . . , vn)>0otherwise it is negatively oriented.

8. Ifω∈Λⁿ(V) is non-zero thenω(v₁, . . . , v_n)6= 0 if and only if v₁, . . . , v_n is a basis of V.

9. If e1, . . . , en is the standard basis ofRⁿ and φ1, . . . , φn is the dual basis of Λ¹(Rⁿ), show that det = φ1∧. . .∧φn, where det∈Λⁿ(Rⁿ) is the determinant.

10. det is the standard orientation ofRⁿ. Show that forR²andR³it gives the same definition of orientation that we have in physics.

(a) Ifv1, v2 is a orthornormal basis ofR² (with respect to the standard inner product onRⁿ), then it is positively oriented ifv1 has to be rotated by an angle ofπ/2 in counter-clockwise direction to getv2.

(b) If v1, v2, v3 is an orthonormal basis for R³, then it is positively oriented if it satisfies the right hand cork screw rule. (Hint. Use the cross product.)

(c) Notice that ordering of the basis elements is quite crucial for its orientation. e₁, e₂is a positively oriented basis ofR²where ase2, e1is negatively oriented. Show that in generale1, . . . , enis always positively oriented basis for Rⁿ with respect to the standard orientation, and e_σ(1), . . . , e_σ(n) is positively oriented ifσ∈Sn is an even permutation.