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l_notes08

# l_notes08 - Lecture VIII Scalar Fields Cylindrical...

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| | | | | 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 ˆ u be a fixed unit vector and let P be a fixed point. For any point Q such that PQ is parallel to ˆ u , let ˆ f ( P ) F ( Q ) Δ s ( P, Q ) = PQ u . Consider the limit lim Δ s 0 Δ s ( P,Q ) . If this limit exists, · then we denote it df Δ f ds | u,P = lim ˆ Δ s 0 Δ s and call it the directional derivative of f in the direction ˆ 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 ˆ ∂x | P = ds | i,P . Also, the partial derivative with respect to y equals ˆ

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l_notes08 - Lecture VIII Scalar Fields Cylindrical...

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