Concepts of gradient

Concepts of gradient

1. Definition — Nabla and Gradient

The del operator

acting on a scalar field $φ(x,y,z)$ gives the gradient:

It is a vector field whose components are the partials of $φ$.

For $φ(x,y)$ in 2D, $∇φ = φ_x i + φ_y j$.

2. Geometric Meaning

• Direction: $∇φ(P)$ points normal to the level surface $φ = c$ through $P$, towards increasing $φ$ (steepest ascent direction).
• Magnitude:

(see next-but-one topic).

• Zero: $∇φ=0$ at stationary points (maxima/minima/saddle of $φ$).

Level surfaces $φ=c$ have normal $∇φ$.

3. Properties

• Linearity: $∇(aφ+ bψ)= a∇φ+ b∇ψ$.
• Product: $∇(φψ)= φ∇ψ+ ψ∇φ$.
• For $φ = r = √(x²+y²+z²)$:

(unit radial outward).

• For $φ=r^n$: $∇r^n = n r^{n-2} r$; for $φ=1/r$: $∇(1/r)= -r/r³$.

4. Worked Examples

(a) φ= x² + y² + z²: ∇φ =2x i+2y j+2z k =2 r ; |∇φ|=2r =2√φ ; direction radial outward ✓
    Level sphere φ=c has normal 2r at any point.

(b) φ= x² y + z: ∇φ =2x y i + x² j + k ; at (1,2,3): ∇φ=4 i+1 j+1 k ; |∇φ|=√18 ✓

(c) φ= log(x²+y²+z²)= log r²: ∇φ = (2x/r²)i+(2y/r²)j+(2z/r²)k =2r/r² ✓

(d) Find unit normal to surface S: x²+y²+z²=9 at (1,2,2): φ= x²+y²+z²-9, ∇φ=2x i+2y j+2z k ⇒ at point 2i+4j+4k ⇒ unit n= (1 i+2 j+2 k)/3 ✓

(e) Show φ= e^{x} cos y is harmonic? Need ∇² φ=0? Actually Laplacian uses divergence of gradient.

(f) Directional derivative preview: D_u φ = ∇φ·u where u is unit direction.

5. Physical Significance

• Heat flow: $q = -k ∇T$ (hot to cold, Fourier).
• Electrostatics: $E = -∇V$ (field is negative gradient of potential).
• Steepest ascent on topographic map $z=φ(x,y)$ follows $∇φ$.

6. Exam Points

• Define $∇φ$ and compute for $φ= x²+y²+z², x² y, r^{n}, 1/r$ — 5 marks, very frequent.
• Show $∇φ$ is normal to $φ=c$ and find unit normal — 5 marks, very frequent.
• Magnitude as maximum rate of increase — 3-5 marks.
• Second-order: Laplacian $∇²φ = ∇·∇φ = φ_{xx}+φ_{yy}+φ_{zz}$ — 3 marks, frequent (preview).