Chapter 5 Dual space ∗ • The dual space TP M of TP M is the space of linear mappings ! : TP M! R: 2 We will write the action of ! on vP 2 TP M as !(vP ) or sometimes as h! j vP i : Linearity of the mapping ! means !(uP + avP ) = !(uP ) + a!(vP ) ; (5.1) 8uP ; vP 2 TP M and a 2 R : The dual space is a vector space under the operations of vector ad- dition and scalar multiplication defined by a1!1 + a2!2 : vP 7! a1!1(vP ) + a2!2(vP ) : (5.2) ∗ • The elements of TP M are called dual vectors, covectors, cotangent vectors etc. 2 A dual space can be defined for any vector space V as the space of linear mappings V ! R (or V ! C if V is a complex vector space). Example: Vector Dual vector column vectors row vector kets j i bras hφj Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 functions linear functionals, etc.2 c • Given a function on a manifold f : M! R ; every vector at P produces a number, vP (f) 2 R 8vP 2 TP M : Thus f defines a 16 17 covector df ; given by df(vP ) = vP (f) called the differential or gradient of f. 2 Since vP is linear, so is df ; df(vP + awP ) = (vP + awP )(f) = vP (f) + awP (f) (5.3) 8vP ; wP 2 TP M; a 2 R : ∗ Thus df 2 TP M : ∗ Proposition: TP M is also n-dimensional. Proof: Consider a chart ' with coordinate functions xi : Each xi i i is a smooth function x : M! R : then the differentials dx satisfy i @ @ i @ i −1 i dx j = j (x ) = j x ◦ ' = δj : (5.4) @x P @x P @x '(P ) The differentials dxi are covectors, as we already know. So we ∗ have constructed n covectors in TP M : Next consider a linear combi- i nation of these covectors, ! = !idx : If this vanishes, it must vanish on every one of the basis vectors. In other words, @ ! = 0 ) ! j = 0 @x P i @ ) !idx j = 0 @x P i ) !iδj = 0 i:e: !j = 0 : (5.5) So the dxi are linearly independent. Finally, given any covector !; consider the covector λ = ! − ! @ dxi : Then letting this act on a coordinate basis vector, we @xi P get @ λ j @x P @ @ i @ = ! j − ! i dx j @x P @x P @x P @ @ = ! − ! δi = 08j (5.6) Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 j i j @x P @x P c @ So λ vanishes on all vectors, since the j form a basis. Thus @x P i ∗ ∗ the dx span TP M ; so TP M is n-dimensional. 18 Chapter 5. Dual space ∗ Also, as we have just seen, any covector ! 2 TP M can be written as i @ ! = !idx where !i = ! i ; (5.7) @x P so in particular for ! = df ; we get @ @f !i ≡ (df)i = df i = i (5.8) @x P @x '(P ) This justifies the name gradient. It is straightforward to calculate the effect of switching to another overlapping chart, i.e. a coordinate transformation. In a new chart '0 where the coordinates are yi (and the transition functions are thus yi(x)) we can use Eq. (5.8) to write the gradient of yi as i i @y j dy = j dx (5.9) @x P This is the result of coordinate transformations on a basis of covec- tors. @ Since is the dual basis in T M to fdxig, in order @xi P P @ i for i to be the dual basis to fdy g we must have @y P @ @xj @ i = i j (5.10) @y P @y P @x P These formulae can be generalized to arbitrary bases. Given a vector v, it is not meaningful to talk about its dual, but a a a given a basis feag; we can define its dual basis f! g by ! (eb) = δb : We can make a change of bases by a linear transformation, a 0a a b 0 −1 b ! 7! ! = Ab ! ; ea 7! ea = (A )aeb ; (5.11) 0a 0 a with A a non-singular matrix, so that ! (eb) = δb : Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 Given a 1-form λ we can write it in both bases, c a 0 0a 0 a a λ = λa! = λa! = λaAb ! ; (5.12) 0 −1 b from which it follows that λa = (A )aλb : 19 Similarly, if v is a vector, we can write a 0a 0 0a −1 b v = v ea = v ea = v (A )aeb ; (5.13) a a b and it follows that v = Ab v : • Quantities which transform like λa are called covariant, while those transforming like va are called contravariant. 2 Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 c .