Divergence

from 東大1S熱力学

• Divergence
  • Premise: Consider a vector-valued function $\mathbf{F}:=(F_x,F_y,F_z)$ where the components are functions.
    • Wait, what is this? (blu3mo)
    • Can we think of it like an array of lambda expressions? (blu3mo)
    • $F_x$ and others are variables that take x, y, z respectively.
    • So, it’s like there is a vector at each coordinate in three-dimensional space. (blu3mo)
      • I see. (blu3mo)(blu3mo)(blu3mo)
      • Like electric fields, right? (takker)
        • It might be helpful to find other familiar examples of scalar fields or vector fields.
    • This is called a vector field.
      • It feels like a physical field, and it makes sense.
  • Outflow/Inflow
    • Consider the vectors on each face of a cube.
    • Consider the sum of the outflow rates minus the sum of the inflow rates, and
      • If it’s positive, it’s called outflow.
      • If it’s negative, it’s called inflow.
        • I see. (blu3mo)
  • Divergence
    • Let’s think about outflow/inflow more mathematically.
    • https://www.youtube.com/playlist?list=PLDJfzGjtVLHkFl7M_MjP_Y9R_8EQfVlPP
    • I understood it with Yobinori (blu3mo)
    • Divergence in a small rectangular volume $=\left(\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}\right)dxdydz$
      • In this case, $\frac{\partial F_x}{\partial x}$:
        • By multiplying the change in x (dx) by the slope obtained from partial differentiation, we can obtain the change in the F_x vector when x changes.
        • By multiplying the change in area (dydz), we can find the total divergence of the small rectangular volume.
      • The same is done for y and z.
    • Divergence per unit volume = div F $=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$
      • If we divide the divergence in a small rectangular volume by its volume (dxdydz), we get the value per unit volume.
      • This can also be expressed using the Differential Operator as $\nabla \cdot F$.
        • I see, so that’s why we use the dot product here. (blu3mo)(blu3mo)