  == For reactive impedances ==  Moving here until it's right:  Jacobi obtained his theorem by logical reasoning, but the mathematical proof is as follows.  In the diagram opposite, power is being transferred from the source, with voltage  and fixed source impedance , to a load with resistance , resulting in an instantaneous current .   is simply the source voltage divided by the total circuit resistance:  :  The average power  dissipated in the load is the square of the current multiplied by the resistive portion of the load impedance:  :   We could calculate the value of Zload for which this expression is a maximum, but it is easier to calculate the value of Zload for which the denominator   :  is a minimum.  The result will be the same in either case.  Differentiating with respect to Zload:  :  For a maximum or minimum, the first derivative is zero, so  :  or  :  In practical resistive circuits, Zsource and Zload are both positive.  To find out whether this solution is a minimum or a maximum, we must differentiate again:  :  This is positive for positive values of Zsource and Zload, showing that the denominator is a minimum, and the power is therefore a maximum, when Zsource = Zload*. >