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Vector Calculus - Math

CHAPTER 18 Vector CalculusIn this chapter we develop the fundamental theorem of the Calculus in two and three dimensions. Thisbegins with a slight reinterpretation of that theorem. Consider the endpointsa;bof the interval[a;b]fromatobas the boundary of that interval. Then the fundamental theorem, in this form:( )f(b) f(a)=Zbad fdx(x)dx;relates the values of a function at the boundary with the values of its derivative in the interior. Statedthis way, the fundamental theorems of the Vector Calculus (Green s, Stokes and Gauss theorems) arehigher dimensional versions of the same idea. However, in higher dimensions, things are far morecomplex: regions in the plane have curves as boundaries, andfor regions in space, the boundary is asurface, and surfaces in space have curves as boundaries.

CHAPTER 18 Vector Calculus In this chapter we develop the fundamental theorem of the Calculus in two and three dimensions. This begins with a slight reinterpretation of that theorem.

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