- The collection of partial derivative operators is called DEL operator. Hence DEL can be viewed as the derivative in multi dimensional space.
- DEL operator is defined as a vector differential operator.
- A DEL operator is not a vector in itself, but when acts on a scalar function, it becomes a vector.
- Del is not simply a vector; it is a vector operator. Whereas a vector is an quantity with both a magnitude and direction, DEL does not have a precise value for either until it is allowed to operate on something.
- In Cartesian coordinate system Del operator is given as:
This operator is useful or significant in defining
- Gradient of a scalar V (∇ V)
- Divergence of a vector A (∇ . A)
- Curl of a vector A (∇ x A)
- Laplacian of a scalar V (∇^{2} V)
DEL OPERATOR - CYLINDRICAL CO-ORDINATE SYSTEM:
Unit vectors of Cartesian co-ordinate system are related to unit vectors of Cylindrical co-ordinate system as:
a_{x} = a_{ρ} cosφ – a_{φ} sinφ
a_{y} = a_{ρ} sinφ + a_{φ} cosφ
a_{z} =a_{z}
The differential part of x, y in terms of ρ and φ is given as:
Since the del operator is given as:
DEL OPERATOR - SPHERICAL CO-ORDINATE SYSTEM:
Unit vectors of Spherical co-ordinate system are related to unit vectors of Cartesian co-ordinate system as:
a_{x} = sinθ cosφ a_{r} + cosθ cosφ a_{θ} – sinφ a_{φ}
a_{y} = sinθ sinφ a_{r} + cosθ sinφ a_{θ} + cosφ a_{φ}
a_{z} = cosθ a_{r} - sinθ a_{θ}
The differential part of x, y, z in terms of r, θ and φ as:
Substituting the values, we get
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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