I need to study vector (multivariable) calculus for an exam whose syllabus is roughly the following:

  1. Functions of Two or Three Real Variables: Limit, continuity, partial derivatives, differentiability, maxima and minima.

  2. Integral Calculus: Integration as the inverse process of differentiation, definite integrals and their properties, fundamental theorem of calculus. Double and triple integrals, change of order of integration, calculating surface areas and volumes using double integrals, calculating volumes using triple integrals.

  3. Vector Calculus: Scalar and vector fields, gradient, divergence, curl, line integrals, surface integrals, Green, Stokes and Gauss theorems.

I've browsed through some of suggested books posted in similar threads like: Schaum's Vector Analysis, Schaum's Calculus, Div Grad Curl and all that, Hubbard & Hubbard etc. but since I'm self studying I get stuck at various points because they either have some pre requisites or the explanation is too short. I figured out how to calculate div, grad, curl from the Schaum's book but I'm still confused by the multiple integrals, surface integrals and the three theorems.

Could anyone suggest me a complete beginner's textbook in Vector Calculus which covers the above mentioned topics, has minimal pre requisites and has a lot of solved problems to practice? I'm a math undergrad and this is an entrance exam for a master's course.

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    $\begingroup$ Not an answer, but MathInsight has nice pictures and applets for vector calculus: mathinsight.org/#welcome $\endgroup$ – Giuseppe Negro Sep 24 '18 at 8:16
  • $\begingroup$ A standard text for multivariable calculus will usually have a chapter on Vector Calculus. For example: Stewart -- Calculus - Early Transcendentals, 6th Ed (2008), chapter 16. $\endgroup$ – quasi Sep 24 '18 at 8:32

I would suggest Vector Calculus by Marsden and Tromba. The preface says,

"We require only the bare rudiments of matrix algebra, and the necessary concepts are developed in the text. If this course is preceded by a course in linear algebra, the instructor will have no difficulty enhancing the material. However, we do assume a knowledge of the fundamentals of one-variable calculus $-$ the process of differentiation and integration and their geometric and physical meaning as well as a knowledge of the standard functions, such as the trigonometric and exponential functions."

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  • $\begingroup$ Thanks, I've read it for a few hours and I'm liking the explanations and the liberal use of illustrations. It seems to cover almost everything in the syllabus and there's a solutions manual available which is great for self studying, though I'm not sure if it is done by the authors or a third party. $\endgroup$ – Akash Gaur Sep 26 '18 at 10:12
  • $\begingroup$ Half the solutions are in the back already, didn't notice that! $\endgroup$ – Akash Gaur Sep 26 '18 at 10:21

I went with Calculus: Early Transcendentals by James Stewart. Chapter 12 onwards covers everything that I needed.

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