Final answer:
To conclude φ(x) is harmonic given δ(x) ≥ 0 and δ(x) = 0, we infer from Laplace's equation that the second derivative sum must be zero, which along with the non-negativity, confirms φ(x) obeys conditions for being harmonic.
Step-by-step explanation:
To determine if φ(x) is a harmonic function given δ(x) ≥ 0 and the equation δ(x) = Σ [1 to n] φ'(x) ∂²/∂y² = φ'(x)δ = 0, we should start by understanding the definition of a harmonic function. A harmonic function φ(x) satisfies Laplace's equation, which in two dimensions is ∂²φ/∂x² + ∂²φ/∂y² = 0. Our given condition implies that the sum of the second derivatives with respect to y is non-negative and that φ(x) times this sum is zero. Since φ(x) is not identically zero, we conclude that the sum of the second derivatives itself must be zero, which is one of the conditions for φ(x) to be harmonic.