Differential Operator

from Tokyo University 1S Thermodynamics

• Differential Operator
  • The symbol $\nabla f$ represents the gradient of a function f.
  • Unlike $\frac{df}{dx}$, which only includes the partial derivative with respect to x, $\nabla f$ includes all partial derivatives with respect to x, y, and z (blu3mo).
    • The symbol $\nabla$ is defined as $\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)$.
    • Therefore, $\nabla f=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right)$.
      • So, it seems that $\nabla f$ is not a special operator with built-in operations, but rather a constant (blu3mo).
    • When we want a vector that includes all the gradients, we use the gradient operator.
  • Since the result is a vector, it may be more consistent to use bold font (takker).
    • $\nabla f$
    • But it might be troublesome to write it that way.

from Tokyo University 1S Mechanics A

• $\nabla U$ is the gradient.
  • That makes sense.
  • But what kind of operation is this? It’s not a dot product or a cross product.
    • Is it a multiplication of a vector and a matrix?
• $\nabla \times F$ is rotation.
  • I’m not familiar with this.
  • There are some easy-to-understand explanations in VOICEROID videos (takker.icon).
    • Explanation of Maxwell’s Equations and Monopoles - NicoNico Video
    • Learning High School Physics with Yukipo 4-1-5 [grad, div, rot] - NicoNico Video
• $\nabla \cdot E$ is divergence.
  • As mentioned in Divergence,