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*5.5. Suppose x is a coordinate patch such that gi, = 1 and g,2 0. Prove that the u-curves are geodesics. (Such a p...
Differential Geometry Prove that for a coordinate patch x(u,v), where U is the unit normal defined...
Differential Geometry
Prove that for a coordinate patch x(u,v), where U is the unit
normal defined as
, and K is the Gaussian Curvature.
L, V 1,0) (0,1 1,0) We were unable to transcribe this image
Question 4 (Geodesics on surfaces of revolution) Let S be a surface of revolution and consider...
Question 4 (Geodesics on surfaces of revolution) Let S be a surface of revolution and consider for it the parametrization x(u, v) ((v) cos u, p(v) sin u, ^(v) Assume in addition that (a)2 +()21 (a) Prove that a curve a(t) = x(u(t), v(t)) is a geodesic of S if and only if it satisfies dip 1 ü2 dv p dip p(u)2 0, dv where here and in what follows the dot denotes derivative with respect to t 5 marks...
Sec. 4-5 109 Geodesics 4.12 Prove that 1 d(n g) 2 Ou 1 (In g) du2 = r11 122 and 2 Sec. 4-5 109 Geodesics 4.12 Prov...
Sec. 4-5 109 Geodesics 4.12 Prove that 1 d(n g) 2 Ou 1 (In g) du2 = r11 122 and 2
Sec. 4-5 109 Geodesics 4.12 Prove that 1 d(n g) 2 Ou 1 (In g) du2 = r11 122 and 2
9) A group G is called solvable if there is a sequence of subgroups such that each quotient Gi/Gi-1 is abelian. Here Gi...
9) A group G is called solvable if there is a sequence of subgroups such that each quotient Gi/Gi-1 is abelian. Here Gi-1 Gi means Gi-1 is a normal subgroup of Gi. For example, any abelian group is solvable: If G s abelian, take Go f1), Gi- G. Then G1/Go G is abelian and hence G is solvable (a) Show that S3 is solvable Suggestion: Let Go- [l),Gı-(123)), and G2 -G. Here (123)) is the subgroup generated by the 3-cycle...
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0<x<a, 0<t (2') u(0,y, t)-gi(v), u(a,y,t)-89(v) 0 <y<b, o<t (3) Show th...
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0<x<a, 0<t (2') u(0,y, t)-gi(v), u(a,y,t)-89(v) 0 <y<b, o<t (3) Show that the steady-state solution involves the potential equation, and indicate how to solve it.
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0
11. Consider the parabolic coordinate system (u, v) related to the Cartesian coordi- nates (r, y)...
11. Consider the parabolic coordinate system (u, v) related to the Cartesian coordi- nates (r, y) by х — 2иv, y — u? — u? for (и, v) € [0, оо) х [0, оо) 1 u = 1, u 2' (a) Sketch in the ry-plane the curves given u = 2. Then sketch in 1 v = 1, v = 2. Shade in the region R the xy-plane the curves given v = 2' bounded by the curves given by...
1. Suppose that G is a group. Prove the following two statements: a. If x 2...
1. Suppose that G is a group. Prove the following two
statements:
a. If x 2 G then the inverse of x is unique.
b. If x; y; z 2 G then (xyz)?1 = z?1y?1x?1.
1. Suppose that G is a group. Prove the following two statements: a. If x E G then the inverse of x is unique. b. If x,y,z E G then (xyz)-1 = z-ly-1,-1.
You are given that a 4-dimensional pseudo-Riemannian space-time has the interval ds2dudvf (u) dx2 g?(u) dy*, (u, v, x,...
You are given that a 4-dimensional pseudo-Riemannian space-time has the interval ds2dudvf (u) dx2 g?(u) dy*, (u, v, x, y) in terms of the coordinates x^ = (i) Use the standard variational principle 2 ds dt = 0 dt ti to find the r-equation governing the geodesic, with parameter t, between given points t and t2 (ii) Deduce from the x-geodesic equation obtained in (i) that f' T.. T. =. ur f where a prime denotes differentiation with respect to...
1. Suppose that X, X, X, are iid Berwulli(p),0 <p<1. Let U. - x Show that,...
1. Suppose that X, X, X, are iid Berwulli(p),0 <p<1. Let U. - x Show that, U, can be approximated by the N (np, np(1-P) distribution, for large n and fixed <p<1. 2. Suppose that X1, X3, X. are iid N ( 0°). Where and a both assumed to be unknown. Let @ -( a). Find jointly sufficient statistics for .
1) Find the area between the curves f (x) = x+2 and g(x)= +,x=0, x=2
1) Find the area between the curves f (x) = x+2 and g(x)= +,x=0, x=2
Differential Geometry
Prove that for a coordinate patch x(u,v), where U is the unit
normal defined as
, and K is the Gaussian Curvature.
L, V 1,0) (0,1 1,0) We were unable to transcribe this image
Question 4 (Geodesics on surfaces of revolution) Let S be a surface of revolution and consider for it the parametrization x(u, v) ((v) cos u, p(v) sin u, ^(v) Assume in addition that (a)2 +()21 (a) Prove that a curve a(t) = x(u(t), v(t)) is a geodesic of S if and only if it satisfies dip 1 ü2 dv p dip p(u)2 0, dv where here and in what follows the dot denotes derivative with respect to t 5 marks...
Sec. 4-5 109 Geodesics 4.12 Prove that 1 d(n g) 2 Ou 1 (In g) du2 = r11 122 and 2
Sec. 4-5 109 Geodesics 4.12 Prove that 1 d(n g) 2 Ou 1 (In g) du2 = r11 122 and 2
9) A group G is called solvable if there is a sequence of subgroups such that each quotient Gi/Gi-1 is abelian. Here Gi-1 Gi means Gi-1 is a normal subgroup of Gi. For example, any abelian group is solvable: If G s abelian, take Go f1), Gi- G. Then G1/Go G is abelian and hence G is solvable (a) Show that S3 is solvable Suggestion: Let Go- [l),Gı-(123)), and G2 -G. Here (123)) is the subgroup generated by the 3-cycle...
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0<x<a, 0<t (2') u(0,y, t)-gi(v), u(a,y,t)-89(v) 0 <y<b, o<t (3) Show that the steady-state solution involves the potential equation, and indicate how to solve it.
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0
11. Consider the parabolic coordinate system (u, v) related to the Cartesian coordi- nates (r, y) by х — 2иv, y — u? — u? for (и, v) € [0, оо) х [0, оо) 1 u = 1, u 2' (a) Sketch in the ry-plane the curves given u = 2. Then sketch in 1 v = 1, v = 2. Shade in the region R the xy-plane the curves given v = 2' bounded by the curves given by...
1. Suppose that G is a group. Prove the following two
statements:
a. If x 2 G then the inverse of x is unique.
b. If x; y; z 2 G then (xyz)?1 = z?1y?1x?1.
1. Suppose that G is a group. Prove the following two statements: a. If x E G then the inverse of x is unique. b. If x,y,z E G then (xyz)-1 = z-ly-1,-1.
You are given that a 4-dimensional pseudo-Riemannian space-time has the interval ds2dudvf (u) dx2 g?(u) dy*, (u, v, x, y) in terms of the coordinates x^ = (i) Use the standard variational principle 2 ds dt = 0 dt ti to find the r-equation governing the geodesic, with parameter t, between given points t and t2 (ii) Deduce from the x-geodesic equation obtained in (i) that f' T.. T. =. ur f where a prime denotes differentiation with respect to...
1. Suppose that X, X, X, are iid Berwulli(p),0 <p<1. Let U. - x Show that, U, can be approximated by the N (np, np(1-P) distribution, for large n and fixed <p<1. 2. Suppose that X1, X3, X. are iid N ( 0°). Where and a both assumed to be unknown. Let @ -( a). Find jointly sufficient statistics for .
1) Find the area between the curves f (x) = x+2 and g(x)= +,x=0, x=2
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