It seems unnatural to me that it is so often worthwhile to replace physical objects with their Hodge duals. For instance, if the magnetic field is properly thought of as a 2-form and the electric field as a 1-form, then why do they show up in Ampere's and Gauss' as laws as their duals, i.e.

$$ \int_{\partial M} \star \mathcal B^2 = \int \int_M \left( 4\pi j^2 + \frac{\partial \star \mathcal E^1}{\partial t} \right)$$

$$ \int \int_{\partial U} \star \mathcal E^1 = 4 \pi Q_{enc} $$

Similarly, angular momentum is considered nearly everywhere as a pseudo-vector instead of as a 2-form. Do these laws have formulations that do not use Hodge duals? Is this just for the sake of simplicity, since tensors are less familiar to the physics community than vectors?

It seems unnatural to me that it is so often worthwhile to replace physical objects with their Hodge duals. For instance, if the magnetic field is properly thought of as a 2-form and the electric field as a 1-form, then why do they show up in Ampere's and Gauss' as laws as their duals, i.e.

$$ \int_{\partial M} \star \mathcal B^2 = \int \int_M \left( 4\pi j^2 + \frac{\partial \star \mathcal E^1}{\partial t} \right)$$

$$ \int \int_{\partial U} \star \mathcal E^1 = 4 \pi Q_{\mathrm{enc}} $$

Similarly, angular momentum is considered nearly everywhere as a pseudo-vector instead of as a 2-form. Do these laws have formulations that do not use Hodge duals? Is this just for the sake of simplicity, since tensors are less familiar to the physics community than vectors?

It seems unnatural to me that it is so often worthwhile to replace physical objects with their Hodge duals. For instance, if the magnetic field is properly thought of as a 2-form and the electric field as a 1-form, then why do they show up in Ampere's and Gauss' as laws as their duals, i.e.

$$ \int_{\partial M} \star \mathcal B^2 = \int \int_M \left( 4\pi j^2 + \frac{\partial \star \mathcal E^1}{\partial t} \right)$$

$$ \int \int_{\partial U} \star \mathcal E^1 = 4 \pi Q_{enc} $$

Similarly, angular momentum is considered nearly everywhere as a pseudo-vector instead of as a 2-form. Do these laws have formulations that do not use Hodge duals? Is this just for the sake of simplicity, since tensors are less familiar to the physics community?

It seems unnatural to me that it is so often worthwhile to replace physical objects with their Hodge duals. For instance, if the magnetic field is properly thought of as a 2-form and the electric field as a 1-form, then why do they show up in Ampere's and Gauss' as laws as their duals, i.e.

$$ \int_{\partial M} \star \mathcal B^2 = \int \int_M \left( 4\pi j^2 + \frac{\partial \star \mathcal E^1}{\partial t} \right)$$

$$ \int \int_{\partial U} \star \mathcal E^1 = 4 \pi Q_{enc} $$

Similarly, angular momentum is considered nearly everywhere as a pseudo-vector instead of as a 2-form. Do these laws have formulations that do not use Hodge duals? Is this just for the sake of simplicity, since tensors are less familiar to the physics community than vectors?