Question

Determine the divergence of the gradient of the following scalar function; that is find div(∇f) for:

f(x, y, z) = a x² + b y² + c z², use the following values: a=5, b=3, c=2.

Answer 1

The divergence of the gradient of the scalar function f(x, y, z) = ax² + by² + cz² is 20. Simplifying and substituting the values of a, b, and c, we get div(∇f) = 2a + 2b + 2c = 2(5+3+2) = 20.

Step-by-step explanation:

To find the divergence of the gradient of the given scalar function, we first need to find the gradient of the function. The gradient of a scalar function is a vector given by ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z).

In this case, the function is f(x, y, z) = ax² + by² + cz², where a=5, b=3, and c=2.

Taking the partial derivatives, we have ∂f/∂x = 2ax, ∂f/∂y = 2by, and ∂f/∂z = 2cz.

Next, we find the divergence of the gradient, div(∇f), which is given by the sum of the partial derivatives of the components of the gradient vector. So, div(∇f) = ∂(2ax)/∂x + ∂(2by)/∂y + ∂(2cz)/∂z.

Simplifying and substituting the values of a, b, and c, we get div(∇f) = 2a + 2b + 2c = 2(5+3+2)

= 20.