removing a exterior wall below a floor joist of the second floor
You’re lucky, you will not need to add a support floor joist or beam provided: 1) the floor joists run the direction you show, 2) roof joists run the same direction as the floor joists, 3) the remaining second floor wall does not have a window (as shown), 4) the remaining second floor wall has a minimum of 1/2” plywood sheathing or 1/2” OSB board sheathing. 

1) There is no floor load on the wall you want to remove, if the floor joists run the direction you show. (If the first floor joist is more than 8” from the exterior, you’ll need to add a sister joist where the wall is being removed.)

2) There will be no roof load on the wall you want to remove if the roof joists run the same direction as the floor joist. 

3) and 4) The remaining second floor wall is a “giant” beam that only needs to support itself and a portion of the roof wall above. 

However, if any of the issues I’ve outlined above are different, then you’ll need to add a support beam...or at least double (or triple) the floor joist where the the wall is to be removed. 

The existing footing at the existing post is supporting 1/4 the existing floor load. There is not much load (about 3,100 lbs. plus roof load, if any) transferred to the footing, but should be verified. About 18” x 18” should be adequate, unless your soil condition is a swamp.