# Lecture 2 - Mathematics Review (PHYS 301) - E&M 301 Math...

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E&M11E&M 301Math Tool KitLecture 2Douglas BrymanUBC Physics 301September-December, 2016Vector Calculus IIIntegral Theorems2The Divergence TheoremVAdB dAB3Thus, the of a vector is A is the surface area bounding V.Applies to any continuously differentiable vector field .The of within the volume Vcan bFLUeX :AInterpretationBBDivergence BB dAobtained from the the !value ofB onbounding surfaceRemarkable!VdB4ˆA fluid is characterized by a vector in a cube of volume V x, y,. What is the integral of flow across the total surface?Does satisfy the divergence theorem?xxzvvL2 Activity VAddABB
E&M25ˆ1)ˆˆˆxxxxdyddxdzzdydxzxyvvdAˆA fluid is characterized by a vector in a cube of volume V x, y,. What is the integral of flow across the total surface?Does satisfy the divergence theorem?xxzvvSatisfies the divergence theorem.ddxyz vxdydzxyz6Stokes TheoremxcAB dlBdA7"Conservative Field" : 0 dFl-f FUnder what conditions is 0?dFlFrom Stokes Theorem xis 0 if x0 everywhere. This implies that can bewritten as , since x0.cAff  B dlBdABBB80x0So, a field that is the of a scalar point functi"Conservative Field" : Then, from Stoke's Theorem xIf x0,then(since x0)gradientV (i.e. hn tocAVV F dlFdAFFFFdlis conse potenervativtial) e. Conservative FieldsExamples: Coulomb field, Gravity
E&M39Activity: Show that he Coulomb Field Conservative.