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How can I create new rows from an existing DataFrame by grouping by certain fields (in the example "Country" and "Industry") and applying some math to another field (in the example "Field" and "Value")?

Source DataFrame

df = pd.DataFrame('Country': ['USA','USA','USA','USA','USA','USA','Canada','Canada'],
 'Industry': ['Finance', 'Finance', 'Retail', 
 'Retail', 'Energy', 'Energy', 
 'Retail', 'Retail'],
 'Field': ['Import', 'Export','Import', 
 'Export','Import', 'Export',
 'Import', 'Export'],
 'Value': [100, 50, 80, 10, 20, 5, 30, 10])

 Country Industry Field Value
0 USA Finance Import 100
1 USA Finance Export 50
2 USA Retail Import 80
3 USA Retail Export 10
4 USA Energy Import 20
5 USA Energy Export 5
6 Canada Retail Import 30
7 Canada Retail Export 10

Target DataFrame

Net = Import - Export

 Country Industry Field Value
0 USA Finance Net 50
1 USA Retail Net 70
2 USA Energy Net 15
3 Canada Retail Net 20

There are quite possibly many ways. Here's one using groupby and unstack:

(df.groupby(['Country', 'Industry', 'Field'], sort=False)['Value']
 .sum()
 .unstack('Field')
 .eval('Import - Export')
 .reset_index(name='Value'))

 Country Industry Value
0 USA Finance 50
1 USA Retail 70
2 USA Energy 15
3 Canada Retail 20

IIUC

df=df.set_index(['Country','Industry'])

Newdf=(df.loc[df.Field=='Export','Value']-df.loc[df.Field=='Import','Value']).reset_index().assign(Field='Net')
Newdf
 Country Industry Value Field
0 USA Finance -50 Net
1 USA Retail -70 Net
2 USA Energy -15 Net
3 Canada Retail -20 Net

pivot_table

df.pivot_table(index=['Country','Industry'],columns='Field',values='Value',aggfunc='sum').
 diff(axis=1).
 dropna(1).
 rename(columns='Import':'Value').
 reset_index()
Out[112]: 
Field Country Industry Value
0 Canada Retail 20.0
1 USA Energy 15.0
2 USA Finance 50.0
3 USA Retail 70.0

You can use Groupby.diff() and after that recreate the Field column and finally use DataFrame.dropna:

df['Value'] = df.groupby(['Country', 'Industry'])['Value'].diff().abs()
df['Field'] = 'Net'
df.dropna(inplace=True)
df.reset_index(drop=True, inplace=True)

print(df)
 Country Industry Field Value
0 USA Finance Net 50.0
1 USA Retail Net 70.0
2 USA Energy Net 15.0
3 Canada Retail Net 20.0

You can do it this way to add those rows to your original dataframe:

df.set_index(['Country','Industry','Field'])
 .unstack()['Value']
 .eval('Net = Import - Export')
 .stack().rename('Value').reset_index()

Output:

 Country Industry Field Value
0 Canada Retail Export 10
1 Canada Retail Import 30
2 Canada Retail Net 20
3 USA Energy Export 5
4 USA Energy Import 20
5 USA Energy Net 15
6 USA Finance Export 50
7 USA Finance Import 100
8 USA Finance Net 50
9 USA Retail Export 10
10 USA Retail Import 80
11 USA Retail Net 70

This answer takes advantage of the fact that pandas puts the group keys in the multiindex of the resulting dataframe. (If there were only one group key, you could use loc.)

>>> s = df.groupby(['Country', 'Industry', 'Field'])['Value'].sum()
>>> s.xs('Import', axis=0, level='Field') - s.xs('Export', axis=0, level='Field')
Country Industry
Canada Retail 20
USA Energy 15
 Finance 50
 Retail 70
Name: Value, dtype: int64