A Question on Density Tensors in General Relativity

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The discussion revolves around a homework problem from Gerard t'Hooft's exercise on the transformation rule of a density tensor in general relativity. The original poster struggles to express tensor notation properly and shares a link to a relevant image for reference. They mention attempting to understand the problem through Geroch's "Geometrical Quantum Mechanics" but find the material confusing and unhelpful. The lack of clarity in the text raises questions about whether they are focusing on the correct section for assistance. Overall, the thread highlights challenges in grasping the concepts of density tensors and their transformations in the context of general relativity.
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Homework Statement


this is a problem of exercise of general relativity by Gerard t Hooft

Problem 7, Transformation rule of a density
http://p13.freep.cn/p.aspx?u=v20_p13_photo_1211140612144629_0.jpg

sry i don't know how to type tensors with upper and lower indices. I can only show it by picture

Homework Equations



questions are in the picture

The Attempt at a Solution



I tried to read Geroch's "Geometrical Quantum Mechanics", (III Quantum Mechanics 14. Densities. Integrals. 15. States), but i still cannot understand it
 
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was i in a wrong section that nobody could help?
 
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