curl and their geometrical and physical significance

curl and their geometrical and physical significance

1. Definition — Curl as Cross Product of $∇$ with $F$

For $F = P i + Q j + R k$:

Expanding determinant:

In 2D, $F= P i+Q j$: $∇×F = (Q_x - P_y) k$ (only $k$ component).

2. Properties

• Linearity: $∇×(aF+bG)=a∇×F+b∇×G$.
• $∇×(φF)= φ∇×F + ∇φ×F$.
• $∇·(∇×F)=0$ (divergence of curl zero).
• $∇×(∇φ)=0$ for any scalar $φ$ with continuous second partials (curl of gradient zero) — uses $φ_{xy}=φ_{yx}$.

3. Geometrical / Physical Significance — Rotation Density

• Local rotation (vorticity): $curl F(P)$ = twice the angular velocity of a tiny paddle wheel placed at $P$ in the flow $F$, with direction along rotation axis via right-hand rule. If paddle doesn't spin, $curl F=0$ there.
• Magnitude: $|∇×F|$ measures strength of swirl at $P$; direction is axis of swirl.
• Irrotational field: $∇×F=0$ everywhere ⇔ $(∇×F)_k=0$ ⇔ $P_y = Q_x$ type etc. — field has no local rotation and (on simply connected domain) is conservative: $F = ∇φ$ for some scalar potential $φ$ (path-independent line integral).
  • Fluid: irrotational flow has no vortices.
  • Electrostatics: $∇×E=0$ for static fields ⇒ $E=-∇V$.
• Solenoidal vs Irrotational — Helmholtz:
  • $∇·F=0$ ⇒ solenoidal (no source).
  • $∇×F=0$ ⇒ irrotational (no swirl).

4. Worked Examples

(a) F= -y i + x j (rigid rotation): P=-y, Q=x ⇒ Q_x=1, P_y=-1 ⇒ (∇×F)_k=2 ⇒ ∇×F=2 k
    Non-zero, uniform rotation; magnitude 2 =2ω, here ω=1 ⇒ correct ✓
    Divergence 0 ⇒ both solenoidal and rotational.

(b) F= x i + y j + z k = r: P=x etc. ⇒ all differences zero ⇒ ∇×r =0 (irrotational, as radial) ✓

(c) F= y i + x j ?? Actually F= y i +... : P=y, Q=x ⇒ Q_x=0? Wait Q=x ⇒ Q_x=1? No Q=x ⇒ Q_x=1? Actually Q=x ⇒ Q_x=1? But P=y ⇒ P_y=1 ⇒ Q_x -P_y=0 ⇒ curl 0 ⇒ irrotational ✓
    Check: F= y i + x j = ∇(x y) since ∇(x y)= y i+ x j → curl zero confirms.

(d) F= y z i+ z x j+ x y k: P=y z, Q=z x, R=x y ⇒ R_y= x, Q_z= x ⇒ first comp 0; P_z= y, R_x= y ⇒ second 0; Q_x= z, P_y= z ⇒ third 0 ⇒ curl 0 ⇒ conservative: F=∇(x y z) ✓

(e) Need to watch: F= x y i + y z j + z x k : P= x y, Q= y z, R= z x ⇒ curl = (0 - z)i + (0 - x?) Actually compute: R_y=0 - Q_z= y? Wait Q_z= y, so R_y -Q_z= -y ; P_z -R_x=0 -z = -z ; Q_x -P_y=0 -x = -x ⇒ curl = -y i -z j -x k ≠0 ✓

(f) Potential: if ∇×F=0 on simply connected D, find φ with F=∇φ by line integral ∫ F·dr.

5. Relation to Gradient and Divergence — Identities

• $∇×(∇φ)=0$ — curl-free ⇔ gradient.
• $∇·(∇×F)=0$ — divergence-free ⇔ curl.
• $∇×(∇×F)= ∇(∇·F) - ∇²F$ (vector Laplacian).

6. Exam Points

• Define and compute $∇×F$ via determinant — 5 marks, very frequent.
• Show $∇×(∇φ)=0$ and $∇·(∇×F)=0$ — 5 marks, frequent.
• Determine if $F$ is irrotational ($∇×F=0$) and find scalar potential $φ$ if so — 7 marks, very frequent.
• Physical meaning as twice angular velocity / vortex strength — 3-5 marks, frequent.
• Distinguish solenoidal vs irrotational — 3 marks.