6 September 2002 Physics 217, Fall 2002 1Today in Physics 217: vector derivatives First derivatives:• Gradient (—)• Divergence (—◊)•Curl (—¥) Second derivatives: the Laplacian (—2) and its relatives Vector-derivative identities: relatives of the chain rule, product rule, etc. Image by Eric Carlen, School of Mathematics, Georgia Institute of Technology()()()22ˆˆ,xy x y x y=−++vxy6 September 2002 Physics 217, Fall 2002 2Differential vector calculusdf/dx provides us with information on how quickly a function of one variable, f(x), changes. For instance, when the argument changes by an infinitesimal amount, from x to x+dx, f changes by df, given byIn three dimensions, the function f will in general be a function of x, y, and z: f(x, y, z). The change in f is equal todfdf dxdx=()()()()ˆˆˆˆˆˆfffdf dx dy dzxyzfffdx dy dzxyzfd∂∂∂ =++ ∂∂∂ ∂∂∂ =++⋅++ ∂∂∂ ≡⋅xyzxyzl∇6 September 2002 Physics 217, Fall 2002 3Differential vector calculus (continued)The vector derivative operator — (“del”)produces a vector when it operates on scalar function f(x,y,z).— is a vector, as we can see from its behavior under coordinate rotations: but its magnitude is not a number: it is an operator. ˆˆˆxyz∂∂∂=++∂∂∂xyz∇()ff′=⋅RI∇∇6 September 2002 Physics 217, Fall 2002 4Differential vector calculus (continued)There are three kinds of vector derivatives, corresponding to the three kinds of multiplication possible with vectors:Gradient, the analogue of multiplication by a scalar.Divergence, like the scalar (dot) product.Curl, which corresponds to the vector (cross) product.f∇⋅v∇×v∇6 September 2002 Physics 217, Fall 2002 5GradientThe result of applying the vector derivative operator on a scalar function fis called the gradient of f:The direction of the gradient points in the direction of maximum increase of f(i.e. “uphill”), and the magnitude of the gradient gives the slope of fin the direction of maximum increase.ˆˆˆffffxyz∂∂∂ =++ ∂∂∂ xyz∇6 September 2002 Physics 217, Fall 2002 6DivergenceThe scalar product of the vector derivative operator and a vector function is called the divergenceof the vector function:The divergence of a vector function is a scalar.What is the divergence?If two objects following the direction specified by the vector function increase their separation, the divergence of the vector function is positive. If their separation decreases, the divergence of the vector function is negative.()ˆˆˆ ˆˆˆyxzxyzvvvvvvvxyz xyz∂∂∂∂∂∂∇⋅ = + + ⋅ + + = + +∂∂∂ ∂∂∂xyz xyz6 September 2002 Physics 217, Fall 2002 7A function with constant divergenceThis function hasImage by Eric Carlen (Georgia Tech). ()ˆˆ,xyxy=+vxyxy2.⋅=v∇6 September 2002 Physics 217, Fall 2002 8CurlThe curl of a vector function v isand is, itself, a vector. (To be precise: if v is a vector function, its curl is a pseudovector function.)What is the curl?The curl of a vector function evaluated at a certain point is a measure of how much the vector function’s direction wraps around that point. If there were nearby objects moving in the direction of the function, they would circulate about that point, if the curl were nonzero. ˆˆˆˆˆˆyyzxzxxyzvvvvvvxyz y z z x x yvvv∂∂ ∂∂∂∂∂∂∂×= = − + − + −  ∂∂∂ ∂ ∂ ∂ ∂ ∂ ∂ xyzvxyz∇6 September 2002 Physics 217, Fall 2002 9A function with constant curlThis function hasImage by Eric Carlen (Georgia Tech).()ˆˆ,xyyx=− +vxyxyˆ2.×=vz∇6 September 2002 Physics 217, Fall 2002 10A function with constant curl and divergenceThe two previous functions had nonzero divergence and zero curl, or vice versa. The sum of the two functions, shown here, has (constant) nonzero divergence andcurl. Image by Eric Carlen (Georgia Tech). ()( )( )ˆˆ,xyxyxy=− ++vxyxy6 September 2002 Physics 217, Fall 2002 11And so on…Here’s one with nonzero, nonconstant divergence and constant curl :Image by Eric Carlen (Georgia Tech).()()()22ˆˆ,xy x y x y=−++vxy()2ˆ2xy⋅= +×=vvz∇∇6 September 2002 Physics 217, Fall 2002 12Visualization of divergence and curlGo to this excellent Web site, from which I borrowed the figures at which we’ve been looking:www.math.gatech.edu/~carlen/2507/notes/vectorCalc/(Done by Georgia Tech math professor Eric Carlen).6 September 2002 Physics 217, Fall 2002 13Why are div and curl important in E&M?Consider the electric field from a point charge, and the magnetic field from a constant current in a long straight wire:Nonzero divergence of E indicates the presence of charge; nonzero curl of B indicates the presence of current. These vector derivatives point to the sourcesof the Eand Bfields.B:Iflows out of pageEq6 September 2002 Physics 217, Fall 2002 14Product rules for vector first derivativesThe following product rules involving the vector product will be used frequently:You’ll also find them on the inside front cover of Griffiths, and will prove some of them yourself in recitation.()() ( ) ( )()()()()()()()()()( )()() ()()fg f g g fff fff f=⋅ +⋅⋅=××+××+⋅ +⋅⋅=⋅+⋅⋅×=⋅×−⋅××=×+×××=⋅ −⋅ + ⋅− ⋅AB A B B A A B B AAAAAB B A A BAAAABBAABABBA∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇6 September 2002 Physics 217, Fall 2002 15Vector second derivativesThere are five possibilities for second derivatives involving —: The divergence of a gradient is called the Laplacian, denoted —2: Soon you’ll be good friends with this operator.()2222222ˆˆˆ ˆˆˆfffffxyz xyzfffxyz∂∂∂∂∂∂∇≡⋅ = + + ⋅ + +∂∂∂ ∂∂∂∂∂∂=++∂∂∂xyz xyz∇∇() ()()() ()ff⋅× ⋅⋅× ××vvv∇∇ ∇ ∇ ∇∇∇∇ ∇ ∇6 September 2002 Physics 217, Fall 2002 16Vector second derivatives (continued) The curl of a gradient is always zero, as you’ll show in this week’s homework: The gradient of a divergence, appear frequently in the equations of fluid mechanics, but it never lasts long in the equations of electrodynamics. The divergence of a curl is always zero, as you’ll