Conditions for a Scalar Field Identity

Here explains a vector identity proof by simplifying both sides of an equation using the vector triple product and gradient product rules, ultimately showing that the identity holds only if the scalar field $\phi$ satisfies Laplace's equation ( $\nabla^2 \phi=0$ ). This means the scalar field must be a harmonic function. This theoretical concept is then made practical and intuitive by an app that visually connects a scalar field to its gradient vector field, allowing users to interactively explore these principles and identify harmonic functions.

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✍️Mathematical Proof

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Vector Identity Proof : The problem demonstrates how to prove a vector identity by expanding both sides of the equation, the Left-Hand Side (LHS) and the Right-Hand Side (RHS), and then simplifying them to find the conditions under which they are equal.
Vector Triple Product :The LHS, $\nabla \times(\nabla \phi \times a)$, is a vector triple product that simplifies to $-a\left(\nabla^2 \phi\right)$ by applying the identity $A \times(B \times C)=B(A \cdot C)-C(A \cdot B)$ and using the property that the divergence of a constant vector is zero ( $\nabla \cdot a=0$ ).
Gradient Product Rule : The RHS, $\nabla(\nabla \phi \cdot a)$, is the gradient of a dot product. It simplifies to 0 because the vector $a$ is constant, making both the terms from the product rule equal to zero.
Laplace's Equation and Harmonic Functions : For the identity to hold true, the simplified LHS must equal the simplified RHS. This leads to the condition $-a\left(\nabla^2 \phi\right)=0$. Since $a$ is a non-zero vector, the scalar quantity $\nabla^2 \phi$ must be zero. This final condition, $\nabla^2 \phi=0$, is known as Laplace's equation, and any scalar field $\phi$ that satisfies it is called a harmonic function.

✍️Mathematical Proof

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🧄Proof and Derivation-2

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