CORRECTIONS TO FIRST AND SECOND PRINTINGS OF Olver, P.J., "Equivalence, Invariants, and Symmetry", Cambridge University Press, Cambridge, 1995 Last modified: May 7, 2007 *** on Back Cover, line 17-18, change prospective geometry *** to projective geometry *** page xv, add Faruk G\"ungor, Oleg Morozov, Jeongoo Cheh, Juha Pohjanpelto, Francis Valiquette *** to acknowledgements *** page 22, line 10, change and all $t,s\in \R$ where the equation is defined. *** to all $t,s\in V$ where $V \subset \R^2$ is a connected open subset of the $(t,s)$ plane containing $(0,0)$ consisting of points where the equation is defined. *** page 39, Example 2.13, change the first two occurrences of ${\rm PSL}(n,\R)$ *** to ${\rm PGL}(n,\R)$. *** Also append to the last sentence ${\rm PSL}(n,\R) = \ro{SL}(n,\R)/\{\pm \Id\}$ is equal to the connected component of ${\rm PGL}(n,\R)$ containing the identity. *** page 51, equation (2.14), change $C_{ij}^k = - C_{ij}^k$ *** to $C_{ji}^k = - C_{ij}^k$ *** page 55, lines 4-5, change $G_H = \{g|gHg^{-1} \subset H\}$ has Lie algebra *** to $G_H = \{g|gHg^{-1} \subset H\}$ is a normal subgroup with Lie algebra *** page 61, line 31, change there is a scalar function $h_{\bf v}(t)$ such that *** to there is a function $h_{\bf v}\colon {\Bbb R}^k \to {\Bbb R}^k$ such that *** page 73, line 9, change \pmatrix{a^{-1}\,da & a^{-1} (a\,db - b\,da)\cr0&1\cr} *** to \pmatrix{a^{-1}\,da & a^{-1}\,db\cr0&0\cr} *** (two changes) *** page 85, line 23, change 1 + t h_{\bf v}(x) + \f2 t^2 {\bf v}(h_{\bf v}) + \cdots *** to 1 + t h_{\bf v}(x) + \f2 t^2 [{\bf v}(h_{\bf v}) + h_{\bf v}^2] + \cdots *** page 87, line 10, change \sigma ([{\bf v},{\bf w}]) = \widehat{\bf w}(\sigma ({\bf v})) - \widehat{\bf v}(\sigma ({\bf w})) *** to \sigma ([{\bf v},{\bf w}]) = \widehat{\bf v}(\sigma ({\bf w})) - \widehat{\bf w}(\sigma ({\bf v})) *** i.e. interchange the two terms *** page 93, lines 226-28, change In order to formulate a general theorem governing the existence of relative invariants for sufficiently regular group actions, we consider the extended group action (3.15) on the bundle $E = M\times U$. The key remark is that there is a one-to-one correspondence between relative invariants of weight $\mu $ and linear absolute invariants of the extended action. Specifically, a linear function $J(x,u) = \sum _{\alpha =1}^n R_\alpha (x) u^\alpha $ is an invariant of the extended group action \eq{28} if and only if the vector-valued function $R(x) = (R_1(x),\ldots ,R_q(x))^T$ is a relative invariant of weight $\mu $. Therefore, we need only produce a sufficient number of \iz{linear} invariants of the extended action. Moreover, if $J(x,u)$ is any invariant of the extended group action, then it is not hard to prove that its linear Taylor polynomial is also an invariant, and hence provides a relative invariant for the multiplier representation. Thus, the only question is how many independent relative invariants can be constructed in this manner. *** to In order to formulate a general theorem governing the existence of relative invariants for sufficiently regular group actions, we consider the extended group action (3.15) on the bundle $E = M\times U$ and its dual version $(x,v)\mapsto (g\cdot x,\mu (g,x)^{-T})$ on the dual bundle $E^{\displaystyle *} =X\times U^{\displaystyle *}$. The key remark is that there is a one-to-one correspondence between relative invariants of weight $\mu $ and linear absolute invariants of the dual action. Specifically, a linear function $J(x,v) = \sum _{\alpha =1}^n R_\alpha (x) v^\alpha $ is an invariant of the dual action on $E^{\displaystyle *} $ if and only if the vector-valued function $R(x) = (R_1(x),\ldots ,R_q(x))^T$ is a relative invariant of weight $\mu $. Therefore, we need only produce a sufficient number of \iz{linear} invariants of the extended action. Moreover, if $J(x,v)$ is any invariant of the extended group action, then it is not hard to prove that its linear Taylor polynomial is also an invariant, and hence provides a relative invariant for the multiplier representation. Thus, the only question is how many independent relative invariants can be constructed in this manner. *** (multiple corrections) *** page 94, lines 26-28, change I do not know a general theorem that counts the number of relative invariants of multiplier representations that do not satisfy the hypotheses of Theorem 3.36 *** to A general theorem that counts the number of relative invariants of multiplier representations in all cases can be found in the recent paper by M. Fels and the author, ``On relative invariants'', {\it Math. Ann.} {\bf 308} (1997) 701--732. *** page 96, (3.30), change \bo v_- = a_1 {\partial \over \partial a_0} + 2 a_2 {\partial \over \partial a_1} + \cdots + (n-1) a_{n-1} {\partial \over \partial a_{n-2}} + n a_n{\partial \over \partial a_{n-1}}, \bo v_0 = - n a_0 {\partial \over \partial a_0} - (n-2) a_1 {\partial \over \partial a_1} + \cdots + (n-2) a_{n-1} {\partial \over \partial a_{n-1}} + n a_n {\partial \over \partial a_n}, \bo v_+ = n a_0 {\partial \over \partial a_1} + (n-1) a_1 {\partial \over \partial a_2} + \cdots + 2 a_{n-2} {\partial \over \partial a_{n-1}} + a_{n-1} {\partial \over \partial a_n}. *** to \bo v_- = n a_1 {\partial \over \partial a_0} + (n-1) a_2 {\partial \over \partial a_1} + \cdots + 2 a_{n-1} {\partial \over \partial a_{n-2}} + a_n{\partial \over \partial a_{n-1}}, \bo v_0 = n a_0 {\partial \over \partial a_0} + (n-2) a_1 {\partial \over \partial a_1} + \cdots + (2-n) a_{n-1} {\partial \over \partial a_{n-1}} - n a_n {\partial \over \partial a_n}, \bo v_+ = a_0 {\partial \over \partial a_1} + 2a_1 {\partial \over \partial a_2} + \cdots + (n-1) a_{n-2} {\partial \over \partial a_{n-1}} + n a_{n-1} {\partial \over \partial a_n}. *** page 108, line 24, change $\cot \theta \ne a$ *** to $\cot t \ne a$ *** page 120, second line after (4.35), change The Lie algebra (4.14) *** to The Lie algebra (4.35) *** page 126, line 12, change $(\Psi^{(n)}) ^{\displaystyle *} \theta $ *** to $\Psi ^{\displaystyle *} \theta $ *** page 142, line 28, change $s_0 = 1$, $s_1 = 2, \ldots , s_{r-3} = s_{r-2} = r-1$ *** to $s_0 = 2$, $s_1 = 3, \ldots , s_{r-3} = s_{r-2} = r-1$ *** (two corrections) *** page 144, line 10, change a^\nu _\mu *** to A^\nu _\mu *** page 148, equation (5.15), change the signs in {\bf v}_0 = x \partial _x - {n\over 2} u \partial _u {\bf v}_+ = x^2 \partial _x - n x u \partial _u *** to {\bf v}_0 = x \partial _x + {n\over 2} u \partial _u {\bf v}_+ = x^2 \partial _x + n x u \partial _u *** page 159, lines 5, 15 & 18, change $d_nK_1\wedge \cdots \wedge d_nK_r $ *** to $d_{n+1}[\CD K_1 ]\wedge \cdots \wedge d_{n+1}[\CD K_r] $ *** page 171, line 20, change $n+2$ *** to $n+1$ *** page 171, line -8, change $n+2$ *** to $n+1$ *** page 171, line -7 to -3, delete sentence Moreover, if the stable ... have order at most $n+1$. *** page 179, Example 5.52, line 2, after "...standard representation", add $(x,y,u)\mapsto (\alpha x+\beta y,\gamma x+\delta y,u)$, where $\alpha \delta -\beta \gamma =1$ *** page 188, line -2, change $log x = h(u/x)$ *** to $log x = h(u/x^m)$ *** page 190, lines 8-9, change $H$-reduced equationsymmetry reduced equation $\Delta /H = 0$ admits the corresponding {\rm normalizer subgroup} $G_H = \{g|g\cdot H\cdot g^{-1} \subset H\}$ as a symmetry group. *** to $H$-reduced equation $\Delta /H = 0$ admits the quotient group $G_H/H$, where $G_H = \{g|g\cdot H\cdot g^{-1} \subset H\}$ is the {\rm normalizer subgroup}, as a symmetry group. *** (multiple corrections) *** page 190, line 18, change \eta \partial _y + \zeta \partial _u + \zeta ^y\partial _{v_y} *** to \eta \partial _y + \zeta \partial _v + \zeta ^y\partial _{v_y} *** page 190, line 22, change {\bf v} = \partial _y *** to {\bf v} = \partial _v *** page 192, formula (6.32), change (1 + u_x)^{3/2} *** to (1 + u_x^2)^{3/2} *** page 192, formula after (6.32), change (1 + \theta_r^2) *** to (1 + r^2 \theta_r^2)^{3/2} *** page 198, line 9, change y = f(x) *** to w = f(x) *** page 198, equation (6.56), change y *** to w *** page 201, equation (6.61), change r - 1 *** which occurs three times, to r - 2 *** page 226, line 6, change P(t,x,u^{(2n)}) *** to R(t,x,u^{(2n)}) *** page 231, lines -4 & -1, change E(\bar L) *** to \bar E(\bar L) *** page 243, lines 18 & 20, change (x,v,v_y,v_{yy},\ldots) *** to (y,v,v_y,v_{yy},\ldots) *** page 293, line 7, change a_4 = 0 *** to a_4 = a_5 = 0 *** page 293, equations (9.30) & (9.32), change \bar a_6 \omega^3 = a_6 \omega^3 *** to \bar a_6 \bar \omega^3 = a_6 \omega^3 *** page 306, line 13, change \sum_k *** to \sum_j *** page 306, equation (10.7), change \sum_{k=1}^r *** to \sum_{j=1}^r *** page 307, line 13, change \widetilde \alpha ^\kappa = \sum_k z^\kappa _j(x)\,\theta ^j *** to \widetilde \alpha ^\kappa = \sum_j z^\kappa _j(x)\,\theta ^j *** page 307, equation (10.7), change \sum_{k = 1}^r z^\kappa_j \,\theta^j *** to \sum_{k = 1}^m z^\kappa_j \,\theta^j *** page 309, equation (10.12), change \sum_{i=1}^p \; z^\kappa _i \, \theta ^i *** to \sum_{i=1}^m \; z^\kappa _i \, \theta ^i *** page 339, line 6, delete first arc length *** page 341, line -3, change I_4 *** to I_5 *** page 349, line -12, change \alpha ^1 - T^1_{12} \theta ^1 \wedge \theta ^2 - T^1_{13} \theta ^1 \wedge \theta ^3 *** to \alpha ^1 - T^1_{12} \theta ^2 - T^1_{13} \theta ^3 *** page 367, line 10, change $M$ *** to $M$ and $\bar M$ *** page 368, equation (11.30), change = T \omega^1\wedge \omega ^2\wedge \omega ^3 = T \Omega. *** to = T \omega^1\wedge \omega ^2\wedge \omega ^3. *** i.e. drop last equality. *** page 372, lines 13-16, change However, I do not know any naturally occurring examples exhibiting this phenomenon, and, moreover, the prolongation procedure to be discussed below will handle this (remote) possibility as well.) *** to However, the prolongation procedure to be discussed below will handle this possibility as well; an example is the equivalence problem for a parabolic evolution equation analyzed in [{\bf 69}].) *** page 375, line 4, change (12.3) *** to (12.1) *** page 394, lines 16 & 21, change (11.6) *** to (11.7) *** page 394, line 22, change vector $S$ *** to matrix $S$ *** page 395, equation (12.52), change \varpi = \alpha + S \theta, or explicitly, \varpi^i = \alpha^i + \sum_{j=1}^m S^i_j \theta^j *** to \varpi = \alpha - S \theta, or explicitly, \varpi^i = \alpha^i - \sum_{j=1}^m S^i_j \theta^j *** page 406, equation (12.73), change Q_p \Dhat_x Q_{pp}6 Q_{uu} *** to Q_p \Dhat_x Q_{pp}+ 6 Q_{uu} *** page 411, lines 12 & 13, change ... = a(x ... and ... = b(x ... *** to ... = - a(x ... and ... = - b(x ... *** page 423, equation (14.4), change \Phi (t,w) *** to \Phi (t,s) *** page 425, lines 3-6, change There is, however, a four-parameter group action obtained by including the additional generator $z\partial _y$, whose associated one-parameter group $(x,y,z) \mapsto (x , y + \mu z,z)$ can be recovered from the previous group transformations by taking commutators. *** to Moreover, one cannot include these vector fields in a finite-dimensional Lie algebra, since $[\bo v_2, \bo v_3] = \bo v_4 = z \partial _y$, $[\bo v_4, \bo v_3] = \bo v_5 = z^2 \partial _y$, and so on, hence the successive commutators span an infinite-dimensional Lie algebra of vector fields. *** pages 425, lines 33-34, change Relative invariants correspond to {\it linear\/} invariants $J(x,u) = R(x)\cdot u= \sum _{\alpha =1}^q R_\alpha (x) u^\alpha $ of the extended action, *** to Relative invariants of the dual action on $E^{\displaystyle *} = X\times U^{\displaystyle *} $ correspond to {\it linear\/} invariants $J(x,u) = \sum _{\alpha =1}^q R_\alpha (x) u^\alpha $ of the extended action, *** page 472, Table 1, Case 1.8, column 3, change \frak a(1) \semidirect {\Bbb C}^k *** to {\Bbb C} \semidirect ({\Bbb C} \semidirect {\Bbb C}^k) *** page 479 in refs [37--38], change Compl\'etes *** to Compl\`etes *** page 479 in refs [37--42], change Gauthiers *** to Gauthier *** page 483, reference [128], change dx/dy *** to dy/dx *** pages 477, 478, 480, 486 & 487, update the following references [{\bf 8}] Anderson, I.M., and Kamran, N., The variational bicomplex for second order scalar partial differential equations in the plane, {\it Duke Math. J.}, to appear. [{\bf 29}] Bryant, R.L., and Griffiths, P.A., Characteristic cohomology of differential systems I, II, {\it J. Amer. Math. Soc.} {\bf 8} (1995), 507--596, {\it Duke Math. J.} {\bf 78} (1995), 531--676. [{\bf 30}] Bryant, R.L., Griffiths, P.A., and Hsu, L.; Hyperbolic exterior differential systems and their conservation laws, Part I, {\it Selecta Math.} {\bf 1} (1995), 21--112. [{\bf 70}] Fels, M., The equivalence problem for systems of second-order ordinary differential equations, {\it Proc. London Math. Soc.} {\bf 71} (1995), 221--240 [{\bf 139}] Komrakov, B., Primitive actions and the Sophus Lie problem, {\sl in}: {\it The Sophus Lie Memorial Conference, Oslo, 1992}, O.A. Laudal and B. Jahren, eds., Scandinavian Univ. Press, Oslo, 1994, pp. 187--269 [{\bf 188}] Olver, P.J., Non-associative local Lie groups, {\it J. Lie Theory} {\bf 6} (1996), 23--51. [{\bf 190}] Olver, P.J., Sapiro, G., and Tannenbaum, A., Invariant geometric evolutions of surfaces and volumetric smoothing, {\it SIAM J. Appl. Math.} {\bf 57} (1997), 176--194. *** page 500, change Galois, E., 3 *** to Galois, E., 4 *** page 501, change Morikawa, H., 217, [{\bf 170}--{\bf 172}] *** to Morikawa, H., 217, [{\bf 170}] Morrey, C.B., Jr., 346, [{\bf 171}] Mostow, G.D., 41, 61, [{\bf 172}] *** page 504, change affine-invariant arc length, 339 *** to affine-invariant arc length, 241, 339