rejected gradient front end

gradient section front end rejected and replaced but saved here (a) Method 1 The gradient of a scalar field f(x) in Cartesian x-space can be defined by the limit of the following equation as dx → 0, grad f dx = f(x+dx) - f(x) = df This equation shows that the gradient points in the direction of the maximum change of f because, in that direction, the dot product is maximum and so df is maximum. For convenience, write grad f = G to emphasize its vector sense, G dx = df // dx arbitrary Select for dx the same dx = dx(n) used in Section 8 (a) and Section 3 (b), dx'(n) = e'n dx'n → dx(n) = en dx'n = dx // no implied sum x'-space x-space Since only one x'-space coordinate is varying (dx'n), it follows that df = (∂f/∂x'n)dx'n = (∂'nf) dx'n // no implied sum where (∂'nf) = ∂'n f(x(x')) x(x') ≡ F-1(x') If f is a tensorial scalar field, so that f'(x') = f(x), one can also write (∂'nf) = ∂'n f '(x') Now, since dx = dx(n)= en dx'n is given in terms of en, and since en em = δmn, it is convenient to expand G on the ei using the expansion shown in Section 7 (s), where the ei are the reciprocal base vectors of Section 6, G = ΣiG'i ei Recall that the expansion coefficients are the covariant components of G in x'-space. Then, G dx = df (ΣiG'i ei) (en dx'n) = (∂'nf) dx'n ΣiG'i (ei en) dx'n = (∂'nf) dx'n G'n dx'n = (∂'nf) dx'n G'n = (∂'nf) Therefore, grad f = G = G'i ei = (∂'if) ei (∂'nf) = ∂'n f(x(x')) If the function f is a tensorial scalar field, this can also be written as grad f = G = G'i ei = (∂'if ') ei (∂'nf ') = ∂'n f '(x') In this case, G = grad f is seen to be a tensorial vector field and the covariant components of that field transform as (∂'nf ') = Rmn (∂nf ). One could of course also write this other Section 7 (s) expansion grad f = G = G'i ei = (∂'if ') ei (∂'nf ') = ∂'n f '(x') and the contravariant components of G = grad f transform as (∂'nf ') = Rmn (∂nf ). Recall that in the Standard Notation, a vector indicated by just G represents both the contravariant vector and its covariant partner. (b) Method 2 The gradient of f is defined in Cartesian space by [gradf ]n ≡ Gn ≡ ∂nf(x) ≡ nf(x) This is equivalent to the definition used in Method 1 since df = Σi∂if dxi = Σi Gidxi = G dx = grad f dx If f is a tensorial scalar field under F, then Gn = ∂nf(x) are covariant vector field components under F. Since the first equation above is then a tensor equation, we know it is covariant in the sense of Section 7 (u). Therefore in x'-space it becomes [gradf ]'n = G'n ≡ ∂'nf'(x') According to Section 7 (s), vector G can be expanded in either of the following ways, G = G1 + G2 +... = ΣnGn where G = Gn // = un = un G = G'1e1 + G'2 e2 +... = ΣnG'n en where en G = G 'n where in this second way, the G'n are the covariant components of G in x'-space. Using the first way, grad f = G = ΣnGn = Σn ∂nf(x) = Σn ( ∂n) f(x) = f(x) where the vector gradient operator is defined as f ≡ Σn ∂nf or ≡ Σn ∂n Using the second way, one finds instead grad f = G = ΣiG'i ei = Σi∂'if'(x') ei = (∂'if') ei and this agrees with the result of Method 1 for tensorial scalar f. One could write this last result as grad f = Σi (ei ∂'i) f'(x') = 'CL f'(x') if one is willing to define a curvilinear vector gradient operator symbol as CL ≡ Σn en ∂n Of course CL ≠ . It seems best to avoid using such a symbol. Comment: The phrase "G'n are the covariant components of G in x'-space" (used several times above) means that, when G is mapped to G' in x'-space, G' = RG, the coefficients of G' are G'n when G' is expanded onto the basis vectors e'n in x'-space. From Section 7 (s) with V → G, here are the two relevant expansions for side by side comparison: G' = G'1 e'1 + G'2 e'2 +... = ΣnG'n e'n where e'n G' = G'n e'n = g'ni e'i G = G'1e1 + G'2e2 +... = Σn G'n en where en G = G'n en = g'ni ei