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-ordinate system	Laplace operator , ∇2t
1.	Cartesian co-ordinates, x,y,z	∂2t/∂x2 +∂2t/∂y2 +∂2t/∂z2
2.	Cylindrical co-ordinates, r,θ,z	(1/r) ∂/∂r(r ∂t/∂r) +(1/r2) ∂2t/∂θ2 +∂2t/∂z2
3.	Spherical co-ordinates, r,θ,φ	(1/r2) ∂/∂r(r2 ∂t/∂r) +(1/r2sin2φ) ∂2t/∂θ2 +(1/r2sinφ) ∂/∂φ(sinφ ∂t/∂φ)

 

DIVERGENCE

Divergence is a scalar function but it is an operator of a vector function. It is defined for a function t (which is a function of x and y) and it can be differentiated with respect to x and y in a unit vector direction at some predefined point. Directional derivative is found at a point with the use of partial derivatives as explained below. Example to determine the directional derivative of function f(x, y) = x ³ y ² + 8 x⁴ y at point (1, −2) in the direction of a unit vector in z direction at (√3/2, 1/2).

The partial derivative of f(x, y) with respect to x at (1,-2) will be

= 3x²y² + 32 x³ y =3 1²(-2)²+32(1)³(-2) = 12- 64= -52

The partial derivative of f(x, y) with respect to y at (1,-2) will be

=2x³y + 8 x⁴ = 2 1³(-2)+ 8(1)⁴= –4+8=4

Then Divergence D at (x, y) and in the unit vector at (√3/2, 1/2) will be = -52(√3/2) + 4(1/2) = 2 – 26√3