T1: 8. Integration on Manifolds

Question 1

Give the relation between a volume element in two coord systems.

Answer 1

d^d x = Jd^d x ̃

Question 2

How do the metric and Jacobian relate in transform from RNC

Answer 2

g ̃=|g ̃_μν |=-J^2

or or J=√(-g)

Question 3

Give the general expression for the Jacobian

Answer 3

J=|(∂x^μ)/(∂x ̃^ν )|

Question 4

Give the coordinate-independent definition of integration over the manifold

Answer 4

int_V f = int_V d^dx sqrt(-g) f(x)