Solution and Proof for a Vector Identity and Divergence Problem

This app is an interactive educational tool that uses a visualizer to demonstrate and verify key vector calculus concepts. It showcases Euler's Homogeneous Function Theorem for vector fields, proving the identity $(x \cdot \nabla) v=n v$ for different homogeneous vector fields. The tool further applies this principle to compute the divergence of a more complex vector expression, simplifying $\nabla \cdot\{x[x \cdot v]\}$ to $(n+4)(x \cdot v)$. By bridging abstract theory with a dynamic, real-time visualization and calculation, the app makes complex mathematical relationships tangible and easy to understand.

<aside> 🧄

✍️Mathematical Proof

$\gg$Mathematical Structures Underlying Physical Laws

$\complement\cdots$Counselor

</aside>

The central point is that for a homogeneous vector field $v$ of degree $n$, a special relationship exists: the directional derivative in the direction of the position vector $x$ is simply the vector field itself scaled by the degree of homogeneity. This is a direct application of Euler's Homogeneous Function Theorem, and it is expressed by the identity:

$$ (x \cdot \nabla) v=n v $$

This identity is powerful because it allows us to simplify a differential operation into a simple algebraic one, which is key to solving more complex problems.

The second part of the analysis shows how the core identity is used to solve a much more complex problem: computing the divergence $\nabla \cdot\{x[x \cdot v]\}$. The solution relies on a chain of established vector calculus rules, specifically the product rule for divergence and the product rule for the gradient of a dot product. By systematically simplifying each term, the analysis leads to a surprisingly clean final result:

$$ \nabla \cdot\{x[x \cdot v]\}=(n+4)(x \cdot v) $$

This demonstrates how foundational theorems like Euler's can be combined with standard vector calculus identities to derive new, elegant relationships in the field.

🎬Demonstration

This app is a teaching tool that visualizes and verifies the Homogeneous Function Theorem for vector fields. It shows how different vector fields behave and proves that the identity $( x \nabla) v=n v$ holds true for each one. The demo lets you see abstract math concepts come to life with real-time calculations.

the Homogeneous Function Theorem for vector fields