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In Theodore Shifrin's book: Multivariable Mathematics, he defines the derivative of a multivariable vector-valued function as follows:View attachment 8503

Lafontaine in his book: An Introduction to Differential Manifolds, defines the derivative of a multivariable vector-valued function slightly differently as follows:View attachment 8504

Although intuitively the definitions look similar, I need help to show (formally and rigorously) that Shifrin's definition implies Lafontaine's definition and vice versa ...

Can someone please (formally and rigorously) demonstrate that the two definitions are equivalent ...

Hope that someone can help ...

Peter

Lafontaine in his book: An Introduction to Differential Manifolds, defines the derivative of a multivariable vector-valued function slightly differently as follows:View attachment 8504

Although intuitively the definitions look similar, I need help to show (formally and rigorously) that Shifrin's definition implies Lafontaine's definition and vice versa ...

Can someone please (formally and rigorously) demonstrate that the two definitions are equivalent ...

Hope that someone can help ...

Peter