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It seems that most texts on differential geometry books tend to take a quantum leap from calculus without refering the latter. Differentials suddenly becomes forms, functions suddenly becomes diffeomorphisms, direction directives suddenly becomes vector fields...

Can someone reference a good text or set of notes that connects the ideas from calculus to differential geometry so that the two subjects are merged together?

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    $\begingroup$ You might take a look at the book From Calculus to Cohomology. They avoid the use of differentiable manifold for half of the book and introduce differential forms in open set in $\mathbb R^m$. That might make your life a bit easier (but there are some algebra in the book... not so sure now) $\endgroup$ – user99914 May 30 '15 at 7:59
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    $\begingroup$ It's interesting that the most complete "answer" to this question is the only one posted as a comment - the Dunning-Kruger effect in action. $\endgroup$ – theage May 30 '15 at 14:53
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    $\begingroup$ @theage Since I'm included in your comment, I can say for sure you are drawing the wrong conclusion. The reason mine is posted as an answer is, I don't like to post answers in comments. $\endgroup$ – muaddib May 31 '15 at 8:22
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    $\begingroup$ you might find supermath.info/AdvancedCalculus13.pdf useful. I try to start by treating multivariate analysis, Jacobians and all that then I spend the latter half on forms and such. Tu and Lee's manifold texts are certainly helpful, perhaps Conlon, or Munkres also should be considered. $\endgroup$ – James S. Cook Jun 11 '15 at 16:25
  • $\begingroup$ @JamesS.Cook Hey thank you for the wonderful reference! Although I didn't know that liberty university had a math department given all the controversies over creationism $\endgroup$ – Shamisen Expert Jun 11 '15 at 20:47
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Calculus on Manifolds - Spivak

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Loring W. Tu's book is probably the most gentle differential geometry text I've seen.

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