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Forms Tour
Forms Star Operator The star operator is used to change the order of a form. For example if the star operator is operated on a one form it turns into a two form. If it is operated on a two form it turns into a one form. That is the star operator turns the form on which it is operating into all the additional forms not included originaly. For example lets take the star (*) of the one form dx. The forms we can add are dy and dz. so *dx = dydz, wich is a two form in the z direction. If the star operator is operated on a three form it gives us a zero form and the star of a zero form is a three form. The reason for this is that a three form has no additional forms to add to it and the zero form can add all the forms. We can use an example of the star operator to show how forms can give us a different understanding of fields then vectors can. In vectors the electric field, E, vector and the electric flux, D, vector are related by D = eE. Where the small e is the permativity constant. So in vector notation D and E are both vectors multiplied by a constant. In forms they are related by D = *eE. E is a one form representing the electric field. Star E then is a two form (tubes) representing the flux that flows between the sheets of the E field. The D form is then more easily visualized as the flux rather than just another vector quantity. If the vector relationship between the magnetic field, H, vector and the magnetic flux, B, vector is B = uH, where u is the permability constant, what do you think the forms reprensentation would be? |