Line, Surface And Volume Integrals - Field Theory.


- The Line integral of Vector A along a path L is given as

∫L A .dl

- The line integral is the dot product of a vector with a specified curve C.

- We can also say that line integral is the integral of the tangential component of vector A along the curve C.

- If the path of integration is a closed path, the line integral becomes a closed line integral and is called the circulation of A around C.





-  Line Integral is useful in finding the electric field intensity along a path L.

- The surface integral of a vector B across a surface S is defined as

∫s B .ds

- When the surface S is closed, the surface integral becomes the net outward flux of B across S, i.e.







- Surface integral is useful in finding the magnetic flux through a surface S.

- The volume integral of a scalar T over a volume v is given as

∫v T . dv


SOLVED EXAMPLES/NUMERICALS:

Q.1 Calculate the circulation of A = ρ cosφ aρ+ z sinφ az around the edge L of the wedge defined by 0 < ρ < 2, 0 < φ < 60o, z = 0 as shown.













A = ρcosφ aρ+z sinφ az






∫1 A .dl = ∫02 (ρcosφ aρ+zsinφ az) dρaρ = ρ2 cosφ / 2 = 4 / 2 = 2 (since φ = 0o)

∫2 A .dl = ∫0π/3 (ρcosφ aρ + zsinφ az) ρdφaφ = 0

∫3 A .dl = ∫20(ρcosφ aρ+zsinφ az) dρaρ = - 4 cosφ / 2

= -1 (since φ = 60o)






Q.2 Given that H = x2 ax + y2 ay, evaluate ∫L H .dl where L is along the curve y =x2 from (0, 0) to (1, 1).
Ans:
∫L H .dl = ∫( x2 ax + y2 ay ) . (dx ax + dy ay + dz az)
= ∫( x2 dx+ y2 dy )

But on L, y = x2 hence dy = 2x dx

Therefore
∫L H .dl = ∫01[x2 dx+ x4 (2xdx) ] = ∫01( x2 dx+ 2x5 dx )

= | x3/3 |01 + 2 | x6 /6 |01 = 0.667



Q.3 Given that ρs = x2 + xy, calculate ∫s ρsds over the region y ≤ x2, 0< x< 1.
Ans:

∫sρsds = ∫ ∫ x2 dxdy + ∫ ∫ xy dx dy

= ∫01x2 dx ∫dy + ∫01x dx ∫ydy

= ∫01x2 dx | y | + ∫01x dx | y2 / 2 |

= ∫01x4 dx + ∫01 ( x5 / 2) dx = 1/5 + 1/12 = 0.2833



ALSO READ:

- Line , Surface and Volume Intergral.

- Del Operator - Definition and Significance.

- Gradient Of a Scalar (∇ V).

- Numericals / Solved Examples - Gradient Of a Scalar.

- Divergence Of a Vector ( ∇ . A ).

- Numericals / Solved Examples - Divergence Of a Vector.

- Curl Of a Vector ( ∇ x A).

- Laplacian Of a Scalar ( ∇2 V).


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