Lecture on the Fundamental Theorem of Line Integrals (Section 16.3)

Overview

• Review of the Fundamental Theorem of Calculus Part 2 (FTC2)
• Generalize FTC2 to the Fundamental Theorem for Line Integrals
• Discussion on Independence of Path
• Introduction to Conservative Vector Fields
• Characteristics and computation of Conservative Vector Fields
• Relation to Conservation of Energy

Fundamental Theorem of Calculus Part 2

• Integral of rate of change over interval equals net change over that interval.
• Net Change Theorem: ( \int_{a}^{b} f'(x) , dx = f(b) - f(a) ).
• Interprets the gradient of ( f ) as a 'derivative'._

Fundamental Theorem for Line Integrals

• Definition: Let ( C ) be a smooth curve and ( f ) a differentiable function whose gradient is continuous on ( C ).
• Theorem: The line integral over ( C ) of the gradient of ( f ) equals the difference ( f(B) - f(A) ) where ( A ) and ( B ) are endpoints of ( C ).
• Indicates presence of a conservative vector field.

Independence of Path

• Path Definition: A piecewise smooth curve or a union of such curves.
• Path Independence: Integral over any path between two points is the same irrespective of the path taken.
• Conservative Vector Field: Line integral only depends on endpoints; integrals over closed curves are zero.

Proof of Path Independence

• Independence to Zero: If path independent, integral over any closed path is zero.
• Zero to Independence: If integral over any closed path is zero, path is independent.

Conservative Vector Fields

• Definition: Vector field is conservative if ( F = \nabla f ) for some potential function ( f ).
• Open and Connected Regions: Regions where path independence implies conservativeness.

Simple and Simply Connected Regions

• Simple Curve: Does not intersect itself.
• Simply Connected Region: No holes; can enclose any point with a simple closed curve within the region.

Theorem on Simply Connected Regions

• Theorem: On an open Simply Connected region in ( \mathbb{R}^2 ), if the curl condition is satisfied, the vector field is conservative.

Application to Conservation of Energy

• Work Done: Work by conservative force field is equal to the change in kinetic energy.
• Potential Energy: ( P = -f ).
• Conservation Law: Total energy (kinetic + potential) remains constant.

Examples and Exercises

• Verification of Conservative Fields:
  • Compute partial derivatives to check if a vector field is conservative.
  • Check if the difference in specific partial derivatives is zero for conservativeness.

Computing Conservative Vector Fields

• Potential Function: Find function ( f ) such that ( \nabla f = F ).
• Compare and integrate partial derivatives to build ( f ).

Final Notes

• Proofs and techniques for understanding conservative fields and line integrals.
• Encouraged to explore further into complex analysis and related theorems.