directional derivative

directional derivative

1. Definition

For scalar field $φ(x,y,z)$, unit vector $u = (l,m,n)$ with $l²+m²+n²=1$ giving direction at $P_0$, the directional derivative is:

= rate of change of $φ$ when moving from $P_0$ a small distance $h$ along direction $u$.

2. Formula via Gradient

If $φ$ differentiable at $P_0$:

where $θ$ = angle between $∇φ$ and $u$. Derivation uses chain rule on $φ(r_0+ t u)$.

Consequences:

• Maximum $D_u φ = |∇φ|$ when $u$ is parallel to $∇φ$ ($θ=0$) — direction of steepest increase.
• Minimum $-|∇φ|$ when $u$ opposite $∇φ$ — steepest decrease.
• Zero when $u ⊥ ∇φ$ — moving along level surface $φ=c$, no change (tangent direction).
• If $u$ not unit, divide by $|u|$: $D_u φ = ∇φ·u/|u|$.

3. Working Steps (recipe)

1. Compute $∇φ = (φ_x, φ_y, φ_z)$ at $P_0$.
2. Make direction vector $a = PQ$ (from $P_0$ to $Q$) and unitise $u = a/|a|$.
3. Dot: $D_u φ = ∇φ·u$.

For 2D $φ(x,y)$, same with $u=(cosα, sinα)$.

4. Worked Examples

(a) φ= x² + y² + z² at P(1,2,2) towards Q(2,3,3):
    ∇φ=(2x,2y,2z)=(2,4,4); vector a= Q-P=(1,1,1), |a|=√3 ⇒ u=(1,1,1)/√3
    D_u φ = (2+4+4)/√3 =10/√3 ≈5.77 ; max possible |∇φ|=6 along r direction ✓

(b) φ= x² y z at (1,2,3) along direction i+2j+2k: u=(1,2,2)/3, ∇φ=(2x y z, x² z, x² y)=(12,3,2)
    D= (12·1+3·2+2·2)/3=22/3 ✓

(c) Find direction where φ= x²+ y² decreases fastest at (1,1): ∇φ=(2,2) ⇒ opposite direction (-1,-1)/√2 gives D=-2√2 ✓

(d) Need directional derivative along tangent to curve r(t)? Use u = r'(t)/|r'|.

(e) Temperature T= x²+y²+z²: isotherms spheres; directional derivative along sphere (perpendicular to ∇T) zero ⇒ no temperature change walking on sphere ✓

5. Level Surface Connection

If $u$ lies in tangent plane to $φ=c$ at $P_0$, then $D_u φ=0$ because $u·∇φ=0$ — moving on level surface keeps $φ$ constant to first order.

6. Exam Points

• State $D_u φ = ∇φ·u$ — 3 marks, very frequent.
• Compute $D_u φ$ at $P$ towards $Q$ — 5-7 marks, very frequent.
• Find maximum $D_u φ = |∇φ|$ and its direction $u=∇φ/|∇φ|$ — 5 marks, very frequent.
• Show $D_u φ=0$ for $u$ tangent to $φ=c$ — 3 marks.
• Second-order? Sometimes ask for rate of change of $F$ along curve $r(t)$: $dF/dt = ∇F·r'(t)$ — 3 marks.