Vector Calculus(1st Edition)
Authors:
Durgaprasanna Bhattacharyya
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ISBN: 139796930X, 978-1397969309
Book publisher: Forgotten Books
Book Price $0 : Excerpt From Vector CalculusAnalysis Have Been Adopted From 'the Simplest Physical Ideas Which Immediately Identify Them. Thus The Idea Of Divergence Is Taken Directly From Hydrodynamics, And Keeping Before Our Minds The Picture Of ?uid Leaving (or Entering) A Small Closed Space, We Define The Divergence Of A Vector Function At A Point As The Limit Of The Ratio, If One Exists, Of The Surface-integral Of The Function Over A Small Closed Space Surrounding The Point To The Volume Enclosed By The Surface, A Unique Limit Being Supposed To Be Reached By The Closed Surface Shrinking Up' To A Point In Any Manner. Again, It Is Found That Some Vector Fields Can Be Specified Completely By The Gradient Of A Scalar Function, So That The Line Integral T Of The Vector Function Along Any Closed Curve In (simply Connected) Space Would Vanish. Thus The Work Done Is Nil Along Any Closed Path In A Conservative Field Of Force. But In Case The Vector Function Cannot Be So Specified, An Expression Of This Negative Quality Of The Function At A Point Is Naturally Sought In Its Now Non-evanescent Line Integral Along A Small Closed (plane) Path Surrounding The Point. The Ratio Of This Line Integral To The Area Enclosed By Our Path Generally Approa Ches A Limit As The Path Shrinks Up To A Point, Independently Of Its Original Form And Of The Manner Of Its Shrinking, But Dep Ending Ou The Orientation Of Its Plane. The Limit Moreover Has Usually A Maximum Value, Subject To The Variation Of This Orientation, And A Vector Of Magnitude Equal To This Maximum Value And Drawn Perpendicular To That Aspect Of The Plane Which Gives Us The Maximum Value Is Called The Curl Of The Original Vector Function.
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