数学代写微积分 | AMS 515 HOMEWORK 2

本次数学代写的主要内容是微积分Calculus相关

1.问题1
令vp∈Tp(X)。显示X中存在一条平滑曲线α,在一定间隔内定义

R中约0使得α(0)= p且
α′(0)= vp

ta1,…,xn(p)+ tan)假设

V = y2 + x3 + x

R2上的isavectorfield,f(x,y)= cos(x)+ 2sin(y),p =(π,π4)。计算V(p)(f)

3.问题3
令Rn具有标准的可微结构。固定一个p∈Rn,并为每个v∈Rn设

提示:在p处选择一个图表(U,φ),写v = ai∂| (ai = v(xi))并考虑α-1(x1(p)+

vp∈Tp(Rn)由下式定义

显示
是Rn与Tp(Rn)的同构

vp =α′(0)α:R→Rn

α(t)= p +电视v􏰀→vp

p∂xipp 2.问题2

4.问题4
令ψ:R3→R2为ψ(x,y,z)=(2x + y−z,x + 3z),令T∈(] R2)∗为T(u,v)= u-5v。

计算回撤(ψ∗ T)(x,y,z)

5.问题5

设φ:R2→R3为φ(u,v)=(u,v,u2 + v2),设g0为R3的欧几里得张量。通过将φ向回拉g0来计算度量g1。

1个

2 AMS 515家庭作业2

6.问题6
计算单变量柯西流形上的Fisher信息矩阵。

AMS 515 HOMEWORK 2

1. Question 1
Let vp ∈ Tp(X). Show that there exists a smooth curve α in X, define on some interval

about 0 in R such that α(0) = p and
α′(0) = vp

ta1,…,xn(p)+tan) Suppose that

V =y2 ∂ +x3 ∂ ∂x ∂y

isavectorfieldonR2,f(x,y)=cos(x)+2sin(y),andp=(π,π4). Compute V (p)(f )

3. Question 3
Let Rn have the standard differentiable structure. Fix a p ∈ Rn and for each v ∈ Rn let

Hint: choose a chart (U, φ) at p, write v = ai ∂ | (ai = v (xi))and consider α−1(x1(p) +

vp ∈ Tp(Rn) be defined by where

Show that
is an isomorphism of Rn to Tp(Rn)

vp = α′(0) α : R → Rn

α(t) = p + tv v 􏰀→ vp

p ∂xip p 2. Question 2

4. Question 4
Let ψ : R3 → R2 be ψ(x,y,z) = (2x+y−z,x+3z) and let T ∈ (]R2)∗ be T(u,v) = u−5v.

Compute the pullback (ψ∗T )(x, y, z)

5. Question 5

Let φ : R2 → R3 be φ(u,v) = (u,v,u2 + v2) and let g0 be the Euclidean metric tensor on R3. Compute the metric g1 by pulling back g0 back by φ.

1

2 AMS 515 HOMEWORK 2

6. Question 6
Compute the Fisher information matrix on the univariate Cauchy manifold.

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