Determine whether the function f(x,y) = e^x siny is harmonic.
Recall that a function f is harmonic is
∇2f = 0
Is that true here?
To determine whether the function f(x, y) = e^x * sin(y) is harmonic, we need to check if it satisfies Laplace's equation:
∇^2 f = ∂^2f/∂x^2 + ∂^2f/∂y^2 = 0
Let's calculate the second partial derivatives of f(x, y) and check if their sum is zero.
First, calculate the first partial derivatives:
∂f/∂x = e^x * sin(y)
∂f/∂y = e^x * cos(y)
Next, calculate the second partial derivatives:
∂^2f/∂x^2 = ∂/∂x (e^x * sin(y)) = e^x * sin(y)
∂^2f/∂y^2 = ∂/∂y (e^x * cos(y)) = -e^x * sin(y)
Finally, add the second partial derivatives:
∂^2f/∂x^2 + ∂^2f/∂y^2 = e^x * sin(y) - e^x * sin(y) = 0
Since the sum of the second partial derivatives is zero, the function f(x, y) = e^x * sin(y) satisfies Laplace's equation and is therefore harmonic.