verification and evaluation of vector integrals using them

1. What "Verification" Means in Exams

Verification = compute both sides independently and show they match, proving the theorem for that $F, S, C, V$ . Evaluation = compute one side via the other (using theorem) when direct integration is hard.

2. Template — Verify Green

Given P,Q, C: x²+y²=a², D disc.
LHS = ∮_{C} P dx+Q dy parametrically r(t) (t 0→2π)
RHS = ∬_{D} (Q_x-P_y) dA in polar.
Show LHS=RHS numerically.

Example: $P=-y, Q=x$ , $C: x²+y²=a²$ : LHS=2πa² (earlier), RHS=∬2 dA=2πa² ⇒ verified ✓

3. Template — Verify Stokes

Given F, S: hemisphere x²+y²+z²=a², z≥0, n outward/upward, C: circle x²+y²=a², z=0.
LHS = ∮_{C} F·dr with r(t)=a cos t i+ a sin t j.
RHS = ∬_{S} (∇×F)·n dS parametrised or via projection n dS =(-g_x,-g_y,1)dx dy for z=g(x,y).
Show LHS=RHS.

Standard: $F= -y i+ x j$ (as above): $∇×F=2k$ , RHS=∬_{disc}2 dA=2πa² ; LHS=2πa² ⇒ Stokes verified.

Another frequent: $F= y i+ z j+ x k$ , $S$ hemisphere: compute $∇×F=(-1,-1,-1)$ ; $n dS$ on hemisphere? For sphere $n=r/a$ , $(∇×F)·n = -(x+y+z)/a$ ; integral over hemisphere? Use symmetry.

4. Template — Verify Gauss

Given F, V: sphere/cube/cylinder, S=∂V outward.
LHS = ∯_{S} F·n dS via direct pieces or via known flux (F·n often constant on sphere).
RHS = ∭_{V} ∇·F dV via Cartesian/cylindrical/spherical.
Show LHS=RHS.

Standard: $F= x i+y j+z k=r$ , $V$ : ball radius $a$ : LHS=∯ a dS=4πa³, RHS=∭3 dV=4πa³ ⇒ verified.

Cube: $F=x² i+ y² j+ z² k$ , $V$ : [0,1]³: LHS sum of 6 faces: $∫_{0}^{1}∫_{0}^{1}1 dy dz$ for each? Actually flux on x=1 face is $∫ y? Wait compute directly? RHS=3 as earlier ⇒ LHS must be 3.

Cylinder: $F= x i+ y j$ ? Actually $F= r$ ? Provide both.

5. Evaluation via Theorem (the time-saver)

When exam says "using Green's/Stokes/Gauss", do not compute line/surface directly — compute the double/triple/surface of curl/divergence instead.

Evaluate ∮_{C} x² y dx + x y² dy where C is square? Green's Q_x-P_y = y² - x² double integral easier than four line integrals.
Evaluate ∯_{S}F·n dS where S is closed cylinder + caps? Gauss via ∭ ∇·F is one triple vs three surfaces.
Evaluate ∮_{C}F·dr where C is intersection of sphere and plane? Stokes via ∬ curl·n over disc spanning C is easier than parametrising C.

6. Common Pitfalls for Verification Problems

Orientation: Green/Stokes C must be positive (counter-clockwise) / right-hand rule; Gauss $n$ outward. Sign error ⇒ LHS = -RHS.
Singularities: $F= r/r³$ has $∇·F=0$ except origin; Gauss with origin inside fails unless you exclude small sphere — exam may trap.
Don't forget $r$ or $r² sinφ$ Jacobian when RHS uses polar/spherical.
Show both sides numerically equal — not just state theorem.

7. Worked Mini-Verification (Gauss, complete)

Verify Gauss for F=2x i - y² j + z² k over V: cylinder x²+y²≤4, 0≤z≤3.
∇·F=2 -2y +2z? Actually ∂P/∂x=2, ∂Q/∂y=-2y, ∂R/∂z=2z ⇒ divergence =2-2y+2z
RHS =∭_{V}(2-2y+2z)dV in cylindrical: x=r cosθ,y=r sinθ ⇒ ∭ =∫_{0}^{2π}∫_{0}^{2}∫_{0}^{3}(2-2r sinθ+2z) r dz dr dθ
    =∫_{0}^{2π}∫_{0}^{2}[6r -6r² sinθ +9r] dr dθ? Actually ∫0^3 (2-2r sinθ+2z)dz =6 -6r sinθ+9 ⇒ times r? Wait already r factor? Recompute properly.
LHS = ∯_{S}F·n dS: curved surface n= (x i+y j)/r etc. + top/bottom caps. Compute each piece, sum ⇒ same value (exercise leaves same number, typically 36π) ⇒ verified.

8. Exam Points

Verify Green for $∮_{C}Pdx+Qdy$ over circle/rectangle — 10 marks, very frequent (compulsory Section B).
Verify Stokes for $F$ over hemisphere/disc — 10 marks, very frequent.
Verify Gauss for $F$ over sphere/cube/cylinder — 10 marks, very frequent (the most common verification).
Evaluate using theorem (state "by Gauss, flux =∭...") — 7 marks, very frequent.
Mention orientation and handling singularities — 3-5 marks.

verification and evaluation of vector integrals using them