I need to study vector (multivariable) calculus for an exam whose syllabus is roughly the following:
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Functions of Two or Three Real Variables: Limit, continuity, partial derivatives, differentiability, maxima and minima.
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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.
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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.
Best Answer
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."