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raopeng
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Homework Statement
Let [itex]X^{i}[/itex] be a vector field in Minkowski space [itex]R^{4}_{1}[/itex]. We define the integral of this vector over a 3-dimensional hypersurface as the integral of the 3-form [itex]X^{i}dS_{i}[/itex]. where
[itex]dS_{i}=\frac{1}{6}\sqrt{|g|}ε_{jkli} dx_{j}\lambda dx_{k}\lambda dx_{l}[/itex](don't know how to type exterior derivative)
Prove that:
[itex]\int_{∂V} X^{i}dS_{i} = \int_{V} \frac{\partial X^{i}}{\partial x^{i}}dV[/itex]
From "Modern Geometry, Methods and Applications, Vol I", 26.5 Exercise 2
Homework Equations
Stokes' Equation
The Attempt at a Solution
I thought it was pretty simple so I just simply apply the Stokes' Equation which give me something like this:
[itex]\int_{V} \frac{\partial X^{i}}{\partial x^{i}}\frac{1}{6}\sqrt{|g|}ε_{jkli} ε_{ijkl} dx_{1}\lambda dx_{2}\lambda dx_{3}\lambda dx_{4}[/itex]
while we can calculate the value of ε_{jkli} ε_{ijkl} by substituting indexes and considering its relation to the determinant of δ, the coefficient 1/6 is something I cannot get rid of. And it seems the more general form of this equation can be written as:
[itex]\int_{∂V} X^{i}dS_{i} = \int_{V} ∇_{i} X^{i}dV[/itex] where the coefficient becomes [itex]\frac {1}{(n-1)!}[/itex]. But now I cannot make connections with [itex]\frac {1}{(4-1)!}[/itex]
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