The symbol ∇ (nabla) represents the del operator.
Usage
Definition
∇=(∂x1∂,…,∂xn∂)
For example, using the standard basis i^,j^,k^ of R3, we get
∇=(∂x∂,∂y∂,∂z∂)=i^∂x∂+j^∂y∂+k^∂z∂
Gradient
If f:R3→R is a scalar field,
gradf=∇f=(∂x∂f,∂y∂f,∂z∂f)∈R3
The gradient is the "slope" direction and magnitude.
Divergence
If f:R3→R3 is a vector field and f(x,y,z)=(fx,fy,fz),
divf=∇⋅f=∂x∂fx+∂y∂fy+∂z∂fz∈R
The divergence measures how much field diverges from the given point.
Curl
If f:R3→R3 is a vector field and f(x,y,z)=(fx,fy,fz),
curlf=∇×f=∣∣∣∣∣∣∣∂x∂fxi^∂y∂fyj^∂z∂fzk^∣∣∣∣∣∣∣∈R3
The curl is the torque at a given point.
Directional Derivative
If f:R3→R is a scalar field and a(x,y,z)=(ax,ay,az),
a⋅gradf=(a⋅∇)f=ax∂x∂f+ay∂y∂f+az∂z∂f∈R
Hessian / Laplacian
This one is confusing. ML people use ∇2 to denote the Hessian matrix. For f:R3→R,
∇2f=⎣⎢⎢⎡∂x2∂2f∂y∂x∂2f∂z∂x∂2f∂x∂y∂2f∂y2∂2f∂z∂y∂2f∂x∂z∂2f∂y∂z∂2f∂z2∂2f⎦⎥⎥⎤∈R3
However, in physics, ∇2 denotes the Laplacian operator
Δf=∇2f=∇⋅∇f=∂x2∂2f+∂y2∂2f+∂z2∂2f∈R
Both operators can also be applied on f:R3→Rhigher, but the results will have more dimensions.
References