If : AXY X: AXYx is a coordinate chart, then let k=1k=1. Thus for any (n1,n2,…,nm)(m1,m2,…,xm)in the image of ϕϕ, a path of the kind t↦(x1,x2,…,t+xi,…xm)t↦(x1,x2,…,t+xi,…xm), is transformed, by kk, to a path in MM. Consider the tangent vector, vi(x1,...,xm)vi, at t=0t=0 (x1,...,xm). The vectors v1(x1,...,xm)...vm(x1,...,xm)v1(x1,...,xm)...vm(x1,...,xm) constitute the basis for Tp(X)Tp(M), where p=k(x1,...,xm)p=k(x1,...,xm). As a result, every element in Tp(X)Tp(X) may be expressed as a civicivi combination. This implicitly defines a function cici on Tp(X)Tp(X), for each pp. With this in hand, we can say that the pair is injective superlative.
(b) Replace the suit symbol in B f (f-1 (B)) with ⊂ or ͻ can you say more if f is superlative or injective?
(ϕ,c)(ϕ,c)
Considering a chart (ψ,b)(ψ,b) in TNTN. The mapping, F∗F∗ looks like (F,E)(F,E), where EE = derivative of ϕ∘F∘ψ−1ϕ∘F∘ψ−1, i.e., the function whose value at x=(x1,…,xm)x=(x1,…,xm This is a linear transformation that's the best local approximation of ϕ∘F∘ψ−1ϕ∘F∘ψ−1 at xx. Since FF is smooth, this map is also smooth. Explicitly, assume X=N=RX=N=R, and that the coordinates on XX and NN become called xx and yy. The element Tp(X)Tp(X) is a tangent to RR, and a multiple c[1]pc[1]p of the unit vector [1]p[1]p. Similarly, the description holds for Tq(N)Tq(N). The TMTM coordinates are (x,c)(x,c), and the vector c[1]xc[1]xin TxMTxM is sent to the pair (x,c)∈R2(x,c)∈R2. If f(x)=x2f(x)=x2. Then the element of TMTM whose coords are (x,c)(x,c) is sent, by DFDF, to an element of TNTN whose first coordinate is y=x2y=x2 and whose second coordinate is 2xc2xc. Clearly (x,c)↦(x2,2xc)(x,c)↦(x2,2xc) is hence superlative.
