Line

Line

1. What a Line Integral Is — Two Types

For a curve $C$ with parametrisation $r(t)=x(t)i+y(t)j+z(t)k$, $t∈[a,b]$, and $dr = r'(t)dt$:

• Scalar line integral (with respect to arc length $ds=|r'(t)|dt$):

gives e.g. mass of wire with density $φ$, or length if $φ=1$.

• Vector line integral (work/circulation): for vector field $F = P i+Q j+R k$,

gives work done by $F$ moving along $C$. Flux across $C$ in 2D is $∫_{C} F·n ds$.

2. Properties

• Orientation matters: $∫_{-C} F·dr = -∫_{C} F·dr$ (reverse $t$ flips sign); $∫_{C} φ ds$ independent of direction.
• Additivity: $∫_{C1+C2}=∫_{C1}+∫_{C2}$.
• If $F=∇φ$ (conservative), then:

path independent — test $∇×F=0$ (earlier topic).

3. Worked Examples

(a) F= -y i + x j, C: circle x²+y²=a² counter-clockwise, r(t)=a cos t i+a sin t j, t 0→2π
    F·r' = (-a sin t)(-a sin t)+(a cos t)(a cos t)=a² ⇒ ∫_{C} F·dr =∫_{0}^{2π}a² dt=2πa² ✓
    This is circulation =2·area (since curl=2k).

(b) φ= x², C: line segment from (0,0) to (1,1): r(t)=t i+t j, 0→1, |r'|=√2 ⇒ ∫_{C} φ ds =∫_{0}^{1} t²·√2 dt=√2/3 ✓

(c) F= x i+y j, C: y=x² from (0,0) to (1,1): r(t)=t i+ t² j, r'=(1,2t) ⇒ F·r'= t·1 + t²·2t =t+2t³ ⇒ ∫_{0}^{1}(t+2t³)dt=1 ✓
    Since F=∇(x²+y²)/2 is conservative, check: φ(1,1)-φ(0,0)=1 same.

(d) Closed loop ∮_{C} F·dr : for curl test, use Green's later.

(e) Length of helix r(t)= a cos t i+ a sin t j+ b t k, t 0→2π: length =∫_{0}^{2π} √(a²+b²) dt =2π√(a²+b²) ✓

4. When $F$ Conservative — Short Cut

If $∇×F=0$ on simply connected $D$, find potential $φ$ with $F=∇φ$ by integrating $P=φ_x$ etc., then $∫_{C}F·dr = Δφ$ without parametrising.

5. Exam Points

• Evaluate $∫_{C} F·dr$ for $F= x² i+ y² j$ over circle/line/parabola — 5-7 marks, very frequent.
• Evaluate $∫_{C} φ ds$ and length — 5 marks, frequent.
• Show path independence if $F$ conservative ($∇×F=0$) and compute via potential — 7 marks, very frequent (often asked to choose easier path).