I'm reading the book Introduction to Smooth Manifolds. And there is a question that confuse me on page 202.
Can anyone tell me why it would change under coordinate transformations graphically? Thank you!
2026-05-05 03:45:41.1777952741
Why are are integrals of functions in coordinates on manifolds not invariant under coordinate transformations?
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Hint Consider the coordinate transformation $$f : (x^1, \ldots, x^n) \mapsto (\lambda x^1, \ldots, \lambda x^n) ,$$ where $\lambda > 0$. What is $\operatorname{Vol}(f(C))$?