English in Paper

在英文数学论文中，推导公式时连接前后式子的常用连接词和短语可以分为以下几类，以确保逻辑清晰、推导严谨：

1. 表示直接推导或结果

• 连接词/短语：
  • Therefore
    Example: "Therefore, Equation (3) can be rewritten as..."
    
  • Hence
    Example: "Hence, substituting Equation (2) into Equation (1) gives..."
    
  • Thus
    Example: "Thus, we obtain..."
    
  • Consequently
    Example: "Consequently, the expression simplifies to..."
    
  • It follows that
    Example: "It follows that the solution is..."
    
  • This leads to
    Example: "This leads to the final form of the equation..."

2. 表示代入或替换

• 连接词/短语：
  • By substituting... into...
    Example: "By substituting Equation (4) into Equation (5), we get..."
    
  • Substituting... for...
    Example: "Substituting ( x = y ) for ( z ) in Equation (6) yields..."
    
  • Using
    Example: "Using Equation (7), we can express..."
    
  • Plugging... into...
    Example: "Plugging ( f(x) ) into the integral gives..."

3. 表示简化或变形

• 连接词/短语：
  • Rearranging terms
    Example: "Rearranging terms gives..."
    
  • Simplifying further
    Example: "Simplifying further, we have..."
    
  • Dividing both sides by...
    Example: "Dividing both sides by ( a ), we obtain..."
    
  • Taking the logarithm of both sides
    Example: "Taking the logarithm of both sides, we get..."
    
  • Expanding the expression
    Example: "Expanding the expression leads to..."

4. 表示假设或条件

• 连接词/短语：
  • Assuming that...
    Example: "Assuming that ( f(x) ) is continuous, we derive..."
    
  • Under the assumption that...
    Example: "Under the assumption that ( ), Equation (8) reduces to..."
    
  • Given that...
    Example: "Given that ( x > 0 ), the inequality becomes..."

5. 表示逻辑递进或补充

• 连接词/短语：
  • Furthermore
    Example: "Furthermore, substituting ( y = 0 ) yields..."
    
  • Additionally
    Example: "Additionally, we can generalize this result to..."
    
  • Moreover
    Example: "Moreover, the solution satisfies the boundary condition..."

6. 表示对比或转折

• 连接词/短语（需谨慎使用，避免逻辑混乱）：
  • However
    Example: "However, this result contradicts Equation (9)..."
    
  • In contrast
    Example: "In contrast to Equation (10), the new form shows..."

7. 表示最终结论

• 连接词/短语：
  • Finally
    Example: "Finally, we arrive at the desired result..."
    
  • Thus, the final expression is
    Example: "Thus, the final expression is Equation (11)."