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Divergence and curl intermediate mathematics

Intermediate Mathematics Divergence and Curl R Horan M Lavelle The aim of this package is to provide a short self assessment programme for students who would like to be able to calculate divergences and curls in vector calculus Copyright c rhoran plymouth ac uk mlavelle plymouth ac uk Last Revision Date September Version CTable of Contents Introduction Grad Divergence Div Curl Final Quiz Solutions to Exercises Solutions to Quizzes The full range of these packages and some instructions should they be required can be obtained from our web page Mathematics Support Materials CSection Introduction Grad Introduction Grad The vector di ?erential operator ?? called ??del ? or ??nabla ? is de ?ned in three dimensions to be ? ? ? ?? i j k ? x ? y ? z Note that these are partial derivatives If a scalar function f x y z is de ?ned and di ?erentiable at all points in some region then f is a di ?erentiable scalar ?eld The del vector operator ?? may be applied to scalar ?elds and the result ??f is a vector ?eld It is called the gradient of f see the package on Gradients and Directional Derivatives Quiz As a revision exercise choose the gradient of the scalar ?eld f x y z xy ?? yz a i x ?? z j ?? yk b xyi xyj yk c y i ?? zj ?? yk d y i xy ?? z j ?? yk CSection Introduction Grad The vector operator ?? may also be allowed to act upon vector ?elds Two di ?erent ways in which it may act the subject of this package are extremely important in mathematics science and engineering We will ?rst brie y review some useful properties of vectors Consider the three dimensional vector a a i a j a k We may also write this as a a a a If we multiply it by a constant c then every component of the vector is multiplied by c ca ac ca ca ca If we introduce a second vector b b b b then we recall that there are two di ?erent ways of multiplying vectors together the scalar and vector products The scalar product also called dot product is de ?ned by a b a b a b a b It is a scalar as the name scalar product implies CSection Introduction Grad Quiz Select the scalar product of a and b ?? a b c x ?? y z d The vector product or cross product is de ?ned by a ? b a b ?? a b i ?? a b ?? a b j a b ?? a b k ijk a a a b b b It is a vector as the name vector product implies Note that the second line is a useful shorthand for the ?rst Quiz Choose the vector product of a and b ?? a i ?? j ?? k b ?? i ?? j