物理上的数学方法.pdf

PC4230 Mathematical Methods in Physics Tutorial Exercise 1 1 The n sphere Sn is defi ned to be the set of points in IRn 1satisfying x02 x12 xn2 1 A chart for the region x06 1 is the map x0 x1 xn 7 1 1 x0 x1 xn This is a variation of the stereographic projection that was discussed in the lectures Can you visualise it at least for the cases n 1 2 Consider another chart x0 x1 xn 7 1 1 x0 x1 xn valid for the region x0 6 1 Find the transition functions relating the two charts Hence show that Sn is a diff erentiable manifold 2 a Consider the two maps IR 7 IR3 C 7 3 exp D 7 sin 0 1 sin C and D are curves in IR3 Find the point at which they are tangent b Consider the two functions IR37 IR f x y z 7 x g x y z 7 sinxcosy Find the point at which f and g have the same gradient This example hopefully illustrates the duality between vectors and one forms both correspond to taking the rate of change of a function along a curve In Q3 Q6 assume the manifold is IR3with the standard Cartesian coordinates 3 Draw the one form whose components are 1 0 0 Also draw the one forms whose components are 2 0 0 and 1 0 1 4 Evaluate the following one forms on the tangent vector V 1 2 3 at P 0 2 1 i y2 dx ii z dy y dz iii z2 1 dx dy x2 dz 1 5 a Show that the gradient of a function f can be written as df f x dx f y dy f z dz by acting it on x y and z in turn This provides a rigorous meaning to the rather ill defi ned diff erential df used in elementary calculus b Compute df for each of the following functions i f xy2 yz2 ii f xeyz iii f sin xy cos xz and evaluate it on the vector fi eld V xy x z yz x y 6 A one form is zero at a point P provided V 0 for all tangent vectors V at P A point at which its gradient df is zero is called a critical point of the function f Prove that P is a critical point of f if and only if f x P f y P f z P 0 Find all critical points of f 1 x2 y 1 y2 z In the last two questions assume that the manifold is four dimensional 7 Prove by writing out all the terms that p V V i p ei Let the components of p be 1 1 0 2 those of V be 3 1 1 0 and those of W be 0 2 0 0 Find a p V b p W c p V 3 W and d p V 3 p W 8 a Given four vectors whose components are V 2 1 1 0 W 1 2 0 0 X 0 0 1 1 Y 2 2 0 0 show that they are linearly independent 2 b Find the components of p if p V 1 p W 2 p X 1 p Y 4 c Find the value of p Z for which Z 1 1 0 0 d Determine w