261-F11-sg3

# 261-F11-sg3 - MA 261 Spring 2011 Study Guide 3 You also...

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Unformatted text preview: MA 261 - Spring 2011 Study Guide # 3 You also need Study Guides # 1 and # 2 for the Final Exam 1. Line integral of a function f ( x, y ) along C , parameterized by x = x ( t ) , y = y ( t ) and a ≤ t ≤ b , is ∫ C f ( x, y ) ds = ∫ b a f ( x ( t ) , y ( t )) √ ( dx dt ) 2 + ( dy dt ) 2 dt . (independent of orientation of C , other properties and applications of line integrals of f ) Remarks : (a) ∫ C f ( x, y ) ds is sometimes called the “line integral of f with respect to arc length” (b) ∫ C f ( x, y ) dx = ∫ b a f ( x ( t ) , y ( t )) x ′ ( t ) dt (c) ∫ C f ( x, y ) dy = ∫ b a f ( x ( t ) , y ( t )) y ′ ( t ) dt 2. Line integral of vector field ⃗ F ( x, y ) along C , parameterized by ⃗ r ( t ) and a ≤ t ≤ b , is given by ∫ C ⃗ F · d⃗ r = ∫ b a ⃗ F ( ⃗ r ( t )) · ⃗ r ′ ( t ) dt . (depends on orientation of C , other properties and applications of line integrals of f ) 3. Connection between line integral of vector fields and line integral of functions: ∫ C ⃗ F · d⃗ r = ∫ C ( ⃗ F · ⃗ T ) ds where ⃗ T is the unit tangent vector to the curve C . 4. If ⃗ F ( x, y ) = P ( x, y ) ⃗ i + Q ( x, y ) ⃗ j , then ∫ C ⃗ F · d⃗ r = ∫ C P ( x, y ) dx + Q ( x, y ) dy ; Work = ∫ C ⃗ F · d⃗ r ....
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261-F11-sg3 - MA 261 Spring 2011 Study Guide 3 You also...

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