Hank, W5JFR wrote:
"Can someone please explain including the math, on how a ~500pf
capacitor transforms 10 ohms to 50 ohms for 40 meters?"

Go back to the books and look at L-networks. It`s in the book.

In my 1970, 12th edition of the ARRL Antenna Book, an L-network example
is given for matching a mobile whip to a 50-ohm line. It says on page
296:
"For example, if it is desired to match a 50-ohm liner to a 20-ohm
resistance, Xc will be about 46 ohms, and Xl about 25 ohms."

Xl & Xc come from Fig. 15-12 which is used to match 50-ohm coax to whip
antennas between 10 and 36 ohms.

The Xc of 500pf at 7 MHz is 46 ohms.

Xc = 1 / (6.28)(f)(C)

Best regards, Richard Harrison, KB5WZI