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\Delta \mathcal{J}\equiv \mathcal{J_F}-\mathcal{J_f}=\frac{EI_F}{N}-\frac{EI_f}{n}\approx \frac{1}{n}\int_{[-\frac{L}{2},\frac{L}{2}]^n}\frac{1}{L^n}\cdot \ln\left|\det\left(\frac{\partial_\mathbf{x} f(\mathbf{x})}{\Sigma^{1/2}}\right)\right| d\mathbf{x}-\frac{1}{N}\int_{-[\frac{L}{2},\frac{L}{2}]^N}\frac{1}{L^N}\cdot \ln \left|\det\left(\frac{\partial_\mathbf{X} F(\mathbf{X})}{\Sigma'^{1/2}}\right)\right| dX

\Delta \mathcal{J}\equiv \mathcal{J_F}-\mathcal{J_f}=\frac{EI_F}{N}-\frac{EI_f}{n}\approx \frac{1}{N}\int_{-[\frac{L}{2},\frac{L}{2}]^N}\frac{1}{L^N}\cdot \ln \left|\det\left(\frac{\partial_\mathbf{X} F(\mathbf{X})}{\Sigma'^{1/2}}\right)\right| dX - \frac{1}{n}\int_{[-\frac{L}{2},\frac{L}{2}]^n}\frac{1}{L^n}\cdot \ln\left|\det\left(\frac{\partial_\mathbf{x} f(\mathbf{x})}{\Sigma^{1/2}}\right)\right| d\mathbf{x}

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\Delta \mathcal{J}\approx \frac{1}{n}\mathbb{E}_{\mathbf{x}\sim\mathcal{U}[-\frac{L}{2},\frac{L}{2}]^n}\ln\left|\det\left(\frac{\partial_\mathbf{x} f(\mathbf{x})}{\Sigma^{1/2}}\right)\right|-\frac{1}{N}\mathbb{E}_{X\sim\mathcal{U}[-\frac{L}{2},\frac{L}{2}]^N}\ln \left|\det\left(\frac{\partial_\mathbf{X} F(\mathbf{X})}{\Sigma'^{1/2}}\right)\right|

\Delta \mathcal{J}\approx \frac{1}{N}\mathbb{E}_{X\sim\mathcal{U}[-\frac{L}{2},\frac{L}{2}]^N}\ln \left|\det\left(\frac{\partial_\mathbf{X} F(\mathbf{X})}{\Sigma'^{1/2}}\right)\right| - \frac{1}{n}\mathbb{E}_{\mathbf{x}\sim\mathcal{U}[-\frac{L}{2},\frac{L}{2}]^n}\ln\left|\det\left(\frac{\partial_\mathbf{x} f(\mathbf{x})}{\Sigma^{1/2}}\right)\right|

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