To find the area of each family garden, we need to divide the total area of the field by the number of gardens.
The total area of the field is \( 3.9 , \text{mi}^2 \) and the number of gardens is \( 1.3 \times 10^3 \).
We can perform the division as follows:
\[ \text{Area of each garden} = \frac{3.9 , \text{mi}^2}{1.3 \times 10^3} \]
Calculating that gives:
\[ \text{Area of each garden} = \frac{3.9}{1.3 \times 10^3} \approx 0.0030 , \text{mi}^2 \]
Next, we express \( 0.0030 \) in scientific notation:
\[ 0.0030 = 3.0 \times 10^{-3} \]
Therefore, the area of each garden is:
\[ \boxed{3.0 \times 10^{-3} , \text{mi}^2} \]
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