- #1
Chronum
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https://arxiv.org/pdf/1705.07188.pdf
Equation 5 in this paper states that
$$\frac{\partial F}{\partial p_i} = 2Re\left\lbrace\frac{\partial F}{\partial x}\frac{\partial x}{\partial p_i}\right\rbrace$$
Here, [itex]p_i[/itex] stands for the [itex]i[/itex]'th element of a vector of 'design parameters' [itex]\mathbf{p}[/itex]. These design parameters are variables that we directly control.
Just after that equation, the paper states that the derivative of [itex]x[/itex], involves that 2*Re part because [itex]x[/itex] is complex. [itex]x[/itex] is a vector of complex E and H fields.
Now, my question is why is this? Why is the derivative of complex E and H fields, with respect to a certain parameter [itex]p_i[/itex] cause that extra factor of 2 in the front, and why do we only consider the real part?
Equation 5 in this paper states that
$$\frac{\partial F}{\partial p_i} = 2Re\left\lbrace\frac{\partial F}{\partial x}\frac{\partial x}{\partial p_i}\right\rbrace$$
Here, [itex]p_i[/itex] stands for the [itex]i[/itex]'th element of a vector of 'design parameters' [itex]\mathbf{p}[/itex]. These design parameters are variables that we directly control.
Just after that equation, the paper states that the derivative of [itex]x[/itex], involves that 2*Re part because [itex]x[/itex] is complex. [itex]x[/itex] is a vector of complex E and H fields.
Now, my question is why is this? Why is the derivative of complex E and H fields, with respect to a certain parameter [itex]p_i[/itex] cause that extra factor of 2 in the front, and why do we only consider the real part?