The syllabus for the optional subjects in the BPSC mains exam varies depending on the subject chosen by the candidate. Here is a detailed explanation of the syllabus for some of the popular optional subjects:

1. History:
- Ancient India: Prehistoric period, Indus Valley Civilization, Vedic period, Mauryan Empire, Gupta Empire, etc.
- Medieval India: Delhi Sultanate, Mughal Empire, Maratha Empire, etc.
- Modern India: British rule and Indian freedom struggle.
- World History: Major events and developments.

2. Geography:
- Physical Geography: Earth's structure, landforms, climate, etc.
- Human Geography: Population, migration, urbanization, etc.
- Economic Geography: Agriculture, industries, trade, etc.
- Regional Geography: Study of different regions of the world, including India.

3. Political Science:
- Political Theory: Concepts like democracy, socialism, secularism, etc.
- Indian Government and Politics: Structure and functioning of the Indian government, political parties, elections, etc.
- Comparative Politics: Comparative analysis of different governments and political systems.
- International Relations: Foreign policy, global organizations, etc.

4. Economics:
- Microeconomics: Demand and supply, market structures, etc.
- Macroeconomics: National income, inflation, monetary and fiscal policy, etc.
- Indian Economy: Economic planning, agriculture, industry, etc.
- International Economics: International trade, balance of payments, WTO, etc.

5. Sociology:
- Sociological Concepts and Theories: Functionalism, conflict theory, etc.
- Social Institutions: Family, religion, education, etc.
- Social Change and Development: Social movements, globalization, etc.
- Research Methodology: Methods used in sociological research.

6. Science:
- Physics: Mechanics, optics, electricity, etc.
- Chemistry: Physical, inorganic, and organic chemistry.
- Biology: Cell biology, genetics, ecology, etc.

These are just a few examples of the optional subjects in the BPSC mains exam. The syllabus for each subject is extensive and candidates are required to study the topics in-depth to excel in the examination. It is important to refer to the official BPSC website or the syllabus provided by the commission for the most accurate and up-to-date information.