Basic Insights in Vector Calculus

Basic Insights in Vector Calculus provides an introduction to three famous theorems of vector calculus, Green's theorem, Stokes' theorem and the divergence theorem (also known as Gauss's theorem). Material is presented so that results emerge in a natural way. As in classical physics, we begin with descriptions of flows.The book will be helpful for undergraduates in Science, Technology, Engineering and Mathematics, in programs that require vector calculus. At the same time, it also provides some of the mathematical background essential for more advanced contexts which include, for instance, the physics and engineering of continuous media and fields, axiomatically rigorous vector analysis, and the mathematical theory of differential forms.There is a Supplement on mathematical understanding. The approach invites one to advert to one's own experience in mathematics and, that way, identify elements of understanding that emerge in all levels of learning and teaching.Prerequisites are competence in single-variable calculus. Some familiarity with partial derivatives and the multi-variable chain rule would be helpful. But for the convenience of the reader we review essentials of single- and multi-variable calculus needed for the three main theorems of vector calculus.Carefully developed Problems and Exercises are included, for many of which guidance or hints are provided.Contents: Reviewing Some Calculus EssentialsFluid Motion in 2 Dimensions:2-D Mass-Flow Velocity FieldsCirculation and Green's Theorem 2-D Flux and DivergenceUnifying 2-D ResultsFluid Motion in 3 Dimensions:Flux and DivergenceStokes' TheoremRelative Change in Volumes and in IncrementsSupplement on Mathematical Understanding
Readership: Undergraduate students and lecturers of calculus courses in mathematics major as well as engineering and other related areas.Green's Theorem;Stokes' Theorem;Divergence Theorem;Gauss's Theorem;Flux;Curl;Scalar Curl;Vector Curl;Calculus;Vector Calculus;Fundamental Theorem of Calculus;Mathematical Understanding;Image;Sense;Wonder;Direct Insight;Reflective Insight;Symbolism;Nominal Understanding;Explanatory Understanding;Proof;Syllogism;Pedagogy;Mathematical Development;Genetic Sequences of Understanding;Concept;Conceptual Understanding;Myth;Mathematics Education;Constructivism0Key Features:Suitable for a one-semester first course in undergraduate vector calculusHelps readers get the inside track on three famous theorems, namely, Green's Theorem, Gauss's theorem (Divergence Theorem) and Stokes' TheoremReveals origins in classical physics, without needing students to be in a physics programLeads to key insights that give rise to the three famous theorems of vector calculusPresented so that formulas emerge in a natural wayAccessible to students from numerous STEM majors that require vector calculusProvides background needed in more advanced axiomatically rigorous texts on vector analysis, and applicationsModest prerequisites include basic competence in differential and integral calculus of one real variable; and familiarity with partial derivatives and the chain rule in multi-variable calculusFor the convenience of readers, the book includes a review of immediately relevant aspects of single variable and multi-variable calculusIncludes carefully developed Problems and Exercises, for many of which hints, partial or complete solutions are providedThere is a Supplement on mathematical understanding that invites readers to advert to their own experience and that way discern elements of understanding that emerge in all levels of learning and teaching