MA 227 Standard C12

Fundamental Theorem of Line Integrals

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)