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Unformatted text preview: | | | | | 1 Lecture VIII Scalar Fields Cylindrical Coordinates Scalar Fields Definition 1 Let D be a subset of E 3 . A function f that associates each point P in D to a real number f ( P ) is called a scalar field . D is called the domain of f . Definition 2 Let f be a scalar field on a domain D . Let u be a fixed unit vector and let P be a fixed point. For any point Q such that P Q is parallel to u , let f ( P ) F ( Q ) s ( P, Q ) = P Q u . Consider the limit lim s s ( P,Q ) . If this limit exists, then we denote it df f ds | u,P = lim s 0 s and call it the directional derivative of f in the direction u at P . Definition 3 Let f be a scalar field on D , and let P ( a, b ) be a point in D . The partial derivative of f with respect to x at P is the derivative at a of the function f ( x, b ) . It is denoted by f P . x | The partial derivative with respect to x equals the directional derivative in the i : f df direction x | P...
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