For 2D sphere metric ( ds^2 = R^2(d\theta^2 + \sin^2\theta d\phi^2) ), compute ( R^\theta_\phi\theta\phi ).
Solution:
Nonzero Christoffels: ( \Gamma^\phi_\theta\phi = \cot\theta,\ \Gamma^\theta_\phi\phi = -\sin\theta\cos\theta )
Then ( R^\theta_\phi\theta\phi = \partial_\theta \Gamma^\theta_\phi\phi - \partial_\phi\Gamma^\theta_\phi\theta + ... )
Result: ( R^\theta_\phi\theta\phi = \sin^2\theta )
Scalar curvature ( = 2/R^2 ).
Show that ( T^ij = A^i B^j ) is a tensor of rank (2,0). tensor analysis problems and solutions pdf free
Solution:
Under transformation, ( A'^i = \frac\partial x'^i\partial x^pA^p ), similarly for B. Then
( T'^ij = \frac\partial x'^i\partial x^p\frac\partial x'^j\partial x^q A^p B^q = \frac\partial x'^i\partial x^p\frac\partial x'^j\partial x^q T^pq ) → tensor.
Show that ( \nabla_k g_ij = 0 ) (metric compatibility). For 2D sphere metric ( ds^2 = R^2(d\theta^2
Solution:
Using definition and Christoffel symmetry, proof via substitution.
While many commercial books (e.g., by Schaum’s, Springer, Dover) are not legally free, there are excellent open-access and legally free resources: Show that ( T^ij = A^i B^j ) is a tensor of rank (2,0)
Solve ( \phi ) equation for circular motion ( r = const ).
Solution:
( \ddot\phi = 0 \Rightarrow \phi = \omega t + \phi_0 ), then ( \ddotr = r\omega^2 ) → requires central force.
Simplify: ( \delta_ii )
Solution: ( \delta_ii = \delta_11+\delta_22+\delta_33 = 1+1+1 = 3 ) (in 3D).