Calculus III:

Vector operators
Hat operator
Part operator ∂
Del operator: Gradient and divergence
Curl Operator
Laplace Operator

1. Hat operator

The Hat operator transforms a vector into its unit vetor :

= /||||

The unit vector has a magnitude of 1.

We have || || = 1.

2. Operator part: ∂

The vector partial: ∂ is defined as follows:

∂ = 〈 ∂/∂x, ∂/∂y, ∂/∂z 〉 = 〈 ∂_x, ∂_y, ∂_z 〉

3. Del operator

By definition, It is written as:

∇ = 〈 ∂/∂x, ∂/∂y, ∂/∂z 〉 = ∂/∂x_i

It acts on a scalar to give a vector. This is the gradient of the scalar.

It acts on a vector to give a scalar. This is the divergence of the scalar.

The Gradient:

∇ (scalar) = Vector

∇ φ = 〈 ∂φ/∂x, ∂φ/∂y,∂φ/∂z 〉

The result is a vector.

The Divergence:

∇ . (Vector) = Scalar ( dot product)

∇ . 〈 X, Y, Z 〉 = 〈 ∂/∂x, ∂/∂y, ∂/∂z〉 . 〈 X, Y, Z 〉 =
∂X/∂x + ∂Y/∂y + ∂Z/∂z

∇ . 〈X, Y, Z 〉 = ∂X/∂x + ∂Y/∂y + ∂Z/∂z

The result is a scalar.

4. Curl operator

∇ x (Vector) = Vector (cross product)

∇ x 〈 X, Y, Z 〉 = 〈 ∂/∂x, ∂/∂y, ∂/∂z 〉 x 〈 X, Y, Z 〉 =
〈 ∂Z/∂y - ∂Y/∂z , ∂ X/∂z - ∂Z/∂x , ∂Y/∂x - ∂v/∂y 〉

∇ x ( X, Y, Z) =
〈 ∂Z/∂y - ∂Y/∂z , ∂X/∂z - ∂Z/∂x , ∂Y/∂x - ∂X/∂y 〉

5. Laplacien operator

Δ ( Scalar) = Scalar

Δ = ∇² = ∇ . ∇ =
〈 ∂²/∂x², ∂²/∂y², ∂²/∂z²〉 =
∇ . ∇ ( Scalar) = ∇ . Gradient = Divergence (Gradient)

Δ φ = ∂²φ/∂x² + ∂²φ/∂y² + ∂²φ/∂z²