Multivariable Calculus or Differential Geometry (Analysis on Manifolds) after single variable calculus

Background: Applied Mathematics program, finished with single variable calculus, and in parallel with basic analysis. (Not knowledge of multivariable calculus yet)

Please feel free to recommend multivariable version in the favor of Peter Lax's ?

It is good taste of Peter Lax's new Calculus text, Calculus with Application,for single variable, which balances between theory and application, and more or less a more modern writing with the same spirit to Courant's Introduction to Calculus and Analysis.

Note that I plan to read Zorich's Mathematical Analysis as second reading for Calculus, which contains rigorous development of multivariable calculus. So here the first reading is welcomed, or some comment to convince the level of Zorich is enough for first reading just after single variable calculus.

(Since in some multivariable calculus/analysis textbooks, calculation training is not enough, for example, Duistermaat's Multidimensional Real Analysis does not provide training such as how to compute double integrals, volumes, intuition of directional derivatives etc...)

Or,

Is it ok to directly go to Differential Geometry or analysis on manifolds ?

(Although I am able to compute some double integrals, extreme values, tangent planes according to formula tables, but let's still assume skills more or less zero level, will it provide these computational training with direct entering of differential geometry, e.g. by do Carmo, or analysis of manifolds, e.g. by Munkres or Spivak) ?

• Strong students can use Zorich's book to learn multivariable calculus for the first time. It is intended for students in the first year of university in Russia. Those students only have some limited single-variable calculus from high school. However, it is recommended to use a problem book alongside Zorich, for example the one by Demidovich (which has an English translation). You cannot study differential geometry (manifolds) before studying multivariable calculus and topology (Chapters 7-11 of Zorich). However, Chapters 12-15 of Zorich will mostly be repeated in a differential geometry book. – user208259 Jan 23 '15 at 22:34