Mathematical Methods for Physics and Engineering - Matematica.NET

26.21 ABSOLUTE DERIVATIVES ALONG CURVESthis is the analogue of the expression in Cartesian coordinates discussed insection 26.8.26.21 Absolute derivatives along curvesIn section 26.19 we discussed how to differentiate a general tensor with respectto the coordinates and introduced the covariant derivative. In this section weconsider the slightly different problem of calculating the derivative of a tensoralong a curve r(t) that is parameterised by some variable t.Let us begin by considering the derivative of a vector v along the curve. If weintroduce an arbitrary coordinate system u i with basis vectors e i , i =1, 2, 3, thenwe may write v = v i e i and so obtaindvdt = dvidt e i + v i de idt= dvidt e i + v i ∂e i du k∂u k dt ;here the chain rule has been used to rewrite the last term on the right-hand side.Using (26.75) to write the derivatives of the basis vectors in terms of Christoffelsymbols, we obtaindvdt = dvidt e i +Γ j dukikvi dt e j.Interchanging the dummy indices i and j in the last term, we may factor out thebasis vector and find( )dv dvidt = dt +Γi jkv j duk e i .dtThe term in parentheses is called the absolute (or intrinsic) derivative of thecomponents v i along the curve r(t)and is usually denoted byδv iδt ≡ dvidt +Γi jkv j duk = v i du k; kdt dt .With this notation, we may writedvdt = δviδt e i = v i du k; kdt e i. (26.99)Using the same method, the absolute derivative of the covariant componentsv i of a vector is given byδv iδt ≡ v du ki; kdt .Similarly, the absolute derivatives of the contravariant, mixed and covariant975