Curl Of a Vector Field (Curl A) - Definition, Significance and Solved Examples.

Circulation of a vector field A around a closed path L is defined as:

Mathematically Curl of a vector A is defined as:

Where the area Δ S is bounded by the curve L and the unit vector a_n is the unit vector normal to the surface.


- The direction of the curl is the axis of rotation, as determined by the right hand thumb rule and the magnitude of the curl is the magnitude of rotation.


- From the above relation we can define Curl as the maximum circulation per unit area.


- The curl of a vector field provides another vector field that indicates rotational sources of the original vector field.


- Curl is a measurement of the circulation of vector field A around a particular point.


- If there’s a paper boat in a whirlpool, the circulation would be the amount of force that pushed it along as it went in a circle. The more circulation, the more pushing force you have.


- Consider a closed loop counter C. The circulation will be positive if a component of vector field A is pointing in the direction dl at every point on counter C. Hence if the circulation is positive then obviously the curl of a vector A will also be positive.


- Similarly if a component of vector field A points in the opposite direction (- dl) at every point of the counter, then the circulation and thus the curl will be negative.

- If the curl of a vector field A is zero, then the vector field A is said to be irrotational or potential (if ∇ x A =0). In such cases, the circulation of A around a closed path is zero; it implies that the line integral of A is independent of the chosen path. Hence an irrotational field is also known as a conservative field. 


Curl of a vector A in Cartesian Coordinate system is given as:

The above is the determinant form of the formula for curl. The first line is made up of unit vectors, the second of scalar operators, and the third of scalar functions, so this is not a determinant in the strict mathematical sense.


Curl of vector A in Cylindrical coordinate system is given as:

Curl of vector A in Spherical coordinate system is given as:

STOKES THEOREM:


It states that the circulation of a vector field A around a closed path L is equal to the surface integral of the curl of A over the open surface S bounded by L.

The divergence theorem relates a closed surface integral to an open volume integral, the Stokes theorem relates a closed line integral to an open surface integral.


Q.1 Determine the curl of each of the vector field:
a) A = yz a_x + 4xy a_y + y a_z
Ans:










=> ∇ x A = a_x (1 – 0) + a_y (y – 0) +a_z (4y – z)

= a_x + y a_y + (4y – z) a_z

b) A = ρz sinφ a_ρ + 3ρz²cosφ a_φ
Ans:











=> ∇ x A = a_ρ ( 0 - 6ρzcosφ) + a_φ (ρsinφ – 0) + a_z (6ρz² cosφ – ρzcosφ)

= - 6 ρzcosφ a_ρ + ρsinφ a_φ + (1 / ρ)( 6ρz² cosφ – ρzcosφ) a_z

= - 6 ρzcosφ a_ρ + ρsinφ a_φ + (6z – 1) zcosφ a_z


Q.2 Show that A = (y + z cos xz) a_x + x a_y + xcosxz a_z is conservative, without computing any integrals?
Ans:
If A is conservative, then curl of vector A should be equal to zero.







=>∇ x A = 0.

Hence A is a conservative field.



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).

- Short Notes/FAQ's

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