DEL OPERATOR-LAPLACIAN OPERATOR-DIVERGENCE
DEL OPERATOR-
LAPLACIAN OPERATOR-
DIVERGENCE
These operators help in the study
of fluid mechanics in a much easier
way. These increase the depth of
understanding. Then, it is likely to
be helpful in real life practical applications.
Del operator
∇ = (∂/∂x, ∂/∂y, ∂/∂z)
Gradient is partial derivative of a function with respect to x or y or z
∂p/∂x is pressure gradient in x direction
∂p/∂y is pressure gradient in y direction
∂p/∂z is pressure gradient in z direction
∂v/∂x, ∂v/∂y, and ∂v/∂z are velocity gradients.
∂V/∂x, ∂V/∂y and ∂V/∂z are voltage gradients.
Laplacian operator
The divergence of a gradient of a function is called Laplacian or Laplace operator. It is usually denoted by the symbols ∇·∇ or ∇^{2}, or Δ. It is the second derivative of a function.
Laplacian operator in different common coordinate systems
Sr.No. |
Co-ordinatesystem |
Laplace operator , ∇^{2}t |
1. |
Cartesian co-ordinates, x,y,z |
∂^{2}t/∂x^{2} +∂^{2}t/∂y^{2} +∂^{2}t/∂z^{2} |
2. |
Cylindrical co-ordinates, r,θ,z |
(1/r) ∂/∂r(r ∂t/∂r) +(1/r^{2}) ∂^{2}t/∂θ^{2} +∂^{2}t/∂z^{2} |
3. |
Spherical co-ordinates, r,θ,φ |
(1/r^{2}) ∂/∂r(r^{2} ∂t/∂r) +(1/r^{2}sin^{2}φ) ∂^{2}t/∂θ^{2} +(1/r^{2}sinφ) ∂/∂φ(sinφ ∂t/∂φ) |