The Idea of a Line Integral

Section 12.2 Overview

• Motivating Questions:

  • What is an oriented curve and how is it represented algebraically?
  • Meaning of line integral of a vector-valued function along a curve.
  • Important properties of line integrals.

• Vector Fields:

  • Represent forces like gravity, electromagnetism, or velocities.
  • Dot product calculates work done by constant force along a straight line.
  • More complex when force varies or path isn't a straight line.

Subsection 12.2.1 Orientations of Curves

• Oriented Curve:

  • Direction of travel specified.
  • Different work done depending on path taken.

• Parametrization of Curves:

  • Line segments typically range parameter from start to end.
  • Circles may exchange sine and cosine to adjust orientation.

Subsection 12.2.2 Line Integrals

• Concept of Line Integrals:

  • Break path into smaller pieces.
  • Approximate work done on each piece.
  • Use dot product of force and displacement.

• Riemann Sum Argument:

  • Increase pieces, decrease size → vector field nearly constant.
  • Approximation improves as number of pieces increases.

• Definition 12.2.9:

  • Conditions for limit existence:
    • Smooth curve, piecewise smooth
    • Continuous vector field
  • Line integral = total work by vector field along curve.

Subsection 12.2.3 Properties of Line Integrals

• Integration Over Curves:

  • Break curve into smaller curves.
  • Line integral over entire curve = sum of parts.

• Activity 12.2.4:

  • Determine sign (positive, negative, zero) of line integrals based on vector field interaction.

Subsection 12.2.4 The Circulation of a Vector Field

• Closed Curves & Circulation:
  • Line integral along closed curve = circulation.
  • Measures vector field's tendency to rotate with curve.

Subsection 12.2.5 Summary

• Oriented Curves:

  • Vector-valued function gives orientation.
  • Closed curves end where they start.

• Line Integral:

  • Measures alignment of vector field with curve direction.
  • Analogous properties to definite integrals.

• Circulation:

  • Line integral on closed curve = circulation.

Exercises 12.2.6

• Problems to practice line integrals and their properties.