Der “nabla” oder “del” operator ist ein Vektor mit ∂-derivatives als Komponenten
kartesischer Gradienvektor:
x}i+\frac{∂}{∂ y}j+\frac{∂}{∂ z}k$$ ### Nabla operations interpreted When it acts on a vector field $\vec{v}$, we can form: - **divergence**: $∇⋅\vec{v}$ - **curl**: $\nabla \times \vec{v}$ #### Laplace Operator $$\vec{\nabla}^2 f = \frac{∂^2}{∂ x^2}i+\frac{∂^2}{∂ y^2}j+\frac{∂^2}{∂ z^2}k$$ -> The componentwise [[#laplacian|Laplacian]] of a vector field. $\vec{\nabla}^2 f$ tells you how much $f$ at a point differs from its neighbors ##### Laplacian - If $\nabla^2f>0$, the point is lower than its surroundings (a “valley”). - if $\nabla^2f<0$, the point is higher than its surroundings (a “peak”). - If $\nabla^2f=0$, the function is locally flat or balanced. lisa friend explanation