Preliminary exam prep
Here we consider a map f:M→N between manifolds of dimension m and n, respectively, and the maps that it induces. Let p∈M with x1,…,xm a local chart for U∋p and y1,…,yn a local chart for V∋f(p). Induced from f are the differential (or pushforward) df and the pullback df∗, which are duals of each other:
dfp : TpM→Tf(p)Ndf : TM→TNα↦(β↦α(β∘f))df∗p : T∗f(p)N→T∗pMdf∗ : T∗N→T∗Mω↦ω∘f⋀kT∗N→⋀kT∗Mω dy1∧⋯∧dyk↦(ω∘f) d(y1∘f)∧⋯∧d(yk∘f)
These maps may be described by the diagram below.
Example: For example, consider the map f:R3→R3 given by f(x,y,z)=(x−y,3z2,xz+yz), with the image having coordinates (u,v,w). With elements
2x∂∂x−5z∂∂y∈TM,2uv+√w−5∈C∞(N),cos(uv)∈T∗N,
we have
dfp(2x∂∂x−5z∂∂y)(2uv+√w−5)=(2x∂∂x−5z∂∂y)(6(x−y)z2+√xz+yz−5)(p),df∗p(cos(uv))=cos((x−y)3z2),(⋀2df∗p)(cos(uv)du∧dw)=cos((x−y)3z2)d(3z2)∧d(xz+yz)=cos((x−y)3z2)(−6z2 dx∧dz−6z2 dy∧dz).
dfp : TpM→Tf(p)Ndf : TM→TNα↦(β↦α(β∘f))df∗p : T∗f(p)N→T∗pMdf∗ : T∗N→T∗Mω↦ω∘f⋀kT∗N→⋀kT∗Mω dy1∧⋯∧dyk↦(ω∘f) d(y1∘f)∧⋯∧d(yk∘f)
These maps may be described by the diagram below.
Example: For example, consider the map f:R3→R3 given by f(x,y,z)=(x−y,3z2,xz+yz), with the image having coordinates (u,v,w). With elements
2x∂∂x−5z∂∂y∈TM,2uv+√w−5∈C∞(N),cos(uv)∈T∗N,
we have
dfp(2x∂∂x−5z∂∂y)(2uv+√w−5)=(2x∂∂x−5z∂∂y)(6(x−y)z2+√xz+yz−5)(p),df∗p(cos(uv))=cos((x−y)3z2),(⋀2df∗p)(cos(uv)du∧dw)=cos((x−y)3z2)d(3z2)∧d(xz+yz)=cos((x−y)3z2)(−6z2 dx∧dz−6z2 dy∧dz).
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