Answer:
To find the Laplacian of a function, we need to take the second partial derivatives of the function with respect to each variable and then sum them up. In this case, we have:
f(x, y, z) = e^(x^2 + y^2 + z^2)
Now, let's calculate the Laplacian:
∇^2 f(x, y, z) = ∂^2f/∂x^2 + ∂^2f/∂y^2 + ∂^2f/∂z^2
Taking the partial derivatives:
∂f/∂x = 2xe^(x^2 + y^2 + z^2)
∂^2f/∂x^2 = 2e^(x^2 + y^2 + z^2) + 4x^2e^(x^2 + y^2 + z^2)
∂f/∂y = 2ye^(x^2 + y^2 + z^2)
∂^2f/∂y^2 = 2e^(x^2 + y^2 + z^2) + 4y^2e^(x^2 + y^2 + z^2)
∂f/∂z = 2ze^(x^2 + y^2 + z^2)
∂^2f/∂z^2 = 2e^(x^2 + y^2 + z^2) + 4z^2e^(x^2 + y^2 + z^2)
Now, summing up the second partial derivatives:
∇^2 f(x, y, z) = 2e^(x^2 + y^2 + z^2) + 4x^2e^(x^2 + y^2 + z^2) + 2e^(x^2 + y^2 + z^2) + 4y^2e^(x^2 + y^2 + z^2) + 2e^(x^2 + y^2 + z^2) + 4z^2e^(x^2 + y^2 + z^2)
Simplifying:
∇^2 f(x, y, z) = 8e^(x^2 + y^2 + z^2) + 4x^2e^(x^2 + y^2 + z^2) + 4y^2e^(x^2 + y^2 + z^2) + 4z^2e^(x^2 + y^2 + z^2)
Therefore, the Laplacian of f(x, y, z) = e^(x^2 + y^2 + z^2) is:
∇^2 f(x, y, z) = 8e^(x^2 + y^2 + z^2) + 4x^2e^(x^2 + y^2 + z^2) + 4y^2e^(x^2 + y^2 + z^2) + 4z^2e^(x^2 + y^2 + z^2)
