At the end of the course, each student should be able to…
- C12: FundThmLine. Apply the Fundamental Theorem of Line Integrals.
C12. Fundamental Theorem of Line Integrals
- Due to the Chain Rule
\(\frac{df}{dt}=\nabla f(\vect r(t))\cdot\frac{d\vect r}{dt}\)
and the Fundamental Theorem of Calculus
\(\int_a^b\frac{df}{dt}\,dt=[f]_a^b=f(b)-f(a)\), the line
integral of a gradient field may be computed simply by reversing the
gradient.
- If \(\vect F=\nabla f\), then \(f\) is called a potential function, and \(\vect F\) is called a conservative field.
- The Fundamental Theorem of Line Integrals states that \(\int_C \nabla f\cdot d\vect{r}=[f]_A^B=f(B)-f(A)\) where \(A,B\) are the starting/ending points of the curve \(C\).
- This theorem may be used to find \(\int_C \vect F\cdot d\vect{r}\) more quickly when \(\vect F\) is a conservative field (whenever a potential function \(f\) exists).
- When \(\vect F\) is conservative and \(C\) is a closed loop (it begins and ends at the same point), then \(\int_C\vect F\cdot d\vect{r}=0\).
- The curl of a vector field may be used to determine if a field is
conservative.
- The vector field \(\vect F\) is conservative if and only if \(\curl\vect F=\vect 0\).
- Thus \(\int_C\vect F\cdot d\vect{r}=0\) when \(C\) is a closed loop and \(\curl\vect F=\vect 0\).
Textbook References
- University Calculus: Early Transcendentals (3rd Ed)
- 15.3 (exercises 1-12, 18-24)