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Linear algebra has a lot to say about linear transformations of vectors.

Which field studies non-linear transformations of vector spaces? What is a good introductory textbook on this matter?

Update: Differential geometry might be an answer: studying it.

Multivariable calculus is a mapping from multiple real numbers to a single real number. It is very different from many variables to many variables mapping that a linear algebra does. Complex analysis is a better example, because it captures the mapping of a pair-to-pair of the components of complex numbers. However, Cauchy–Riemann equations restrict derivatives over complex field, allowing a prove very strong theorems.

Is there something less constrained than the complex analysis with many-to-many variable mapping?

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    $\begingroup$ Multivariable calculus, I guess? You may also want to look at differential geometry (after a solid familiarity with multivariable calculus and, perhaps, differential equations has been established) $\endgroup$
    – user562983
    Apr 27, 2019 at 23:19
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    $\begingroup$ Abstract algebra could help. Famous example is SO3 group of rotations in 3D. If you have a sequence of vectors and want to figure out what happens in this sequence then abstract algebra and representation theory will be your friends. $\endgroup$ Apr 29, 2019 at 14:04

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