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AiRAVATA

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I need help solving the following problem:

I know this is silly, but my main problem is that i don't know how the identity [itex]f_{*p}(X(p))=Y_{f(p)}[/itex] looks like. What i mean is the following:

If [itex]X=\sum_{i=1}^m X_i \frac{\partial }{\partial x_i}[/itex], and [itex]Y=\sum_{j=1}^m Y_j \frac{\partial }{\partial y_j}[/itex], then

[tex]f_{*p}(X(p))=f_{*p}(\sum_{i=1}^m X_i(p) \frac{\partial }{\partial x_i})=\sum_{i=1}^m f_{*p}(X_i(p) \frac{\partial }{\partial x_i})[/tex]

so

[tex]f_{*p}(X(p))=\sum_{i=1}^m f_{*p}(X_i(p))\frac{\partial }{\partial x_i}+X_i(p) f_{*p}(\frac{\partial }{\partial x_i})=\sum_{j=1}^m Y_j (f(p))\frac{\partial }{\partial y_j}.[/tex]

Is this correct?

Another question. Does anyone know some good online notes regarding vector bundles?

*Let [itex]M,N[/itex] be differentiable manifolds, and [itex]f\in C^\infty(M,N)[/itex]. We say that the fields [itex]X\in \mathfrak{X}(M)[/itex] and [itex]Y \in \mathfrak{X}(N)[/itex] are f-related if and only if [itex]f_{*p}(X(p))=Y_{f(p)}[/itex] for all [itex]p\in M[/itex].*

Prove that:

(a) [itex]X[/itex] and [itex]Y[/itex] are f-related if and only if [itex]X(g \circ f)=Y(g) \circ f[/itex], for all [itex]g\in C^\infty(M)[/itex].

(b) If [itex]X_i[/itex] is f-related with [itex]Y_i[/itex], [itex]i=1,2[/itex], then [itex][X_1,X_2][/itex] is f-related with [itex][Y_1,Y_2][/itex].

Prove that:

(a) [itex]X[/itex] and [itex]Y[/itex] are f-related if and only if [itex]X(g \circ f)=Y(g) \circ f[/itex], for all [itex]g\in C^\infty(M)[/itex].

(b) If [itex]X_i[/itex] is f-related with [itex]Y_i[/itex], [itex]i=1,2[/itex], then [itex][X_1,X_2][/itex] is f-related with [itex][Y_1,Y_2][/itex].

I know this is silly, but my main problem is that i don't know how the identity [itex]f_{*p}(X(p))=Y_{f(p)}[/itex] looks like. What i mean is the following:

If [itex]X=\sum_{i=1}^m X_i \frac{\partial }{\partial x_i}[/itex], and [itex]Y=\sum_{j=1}^m Y_j \frac{\partial }{\partial y_j}[/itex], then

[tex]f_{*p}(X(p))=f_{*p}(\sum_{i=1}^m X_i(p) \frac{\partial }{\partial x_i})=\sum_{i=1}^m f_{*p}(X_i(p) \frac{\partial }{\partial x_i})[/tex]

so

[tex]f_{*p}(X(p))=\sum_{i=1}^m f_{*p}(X_i(p))\frac{\partial }{\partial x_i}+X_i(p) f_{*p}(\frac{\partial }{\partial x_i})=\sum_{j=1}^m Y_j (f(p))\frac{\partial }{\partial y_j}.[/tex]

Is this correct?

Another question. Does anyone know some good online notes regarding vector bundles?

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