Question

If \(\phi\) is a differentiable scalar function, then div grad \(\phi\) is

• A. 1
• B. \(\nabla \phi\)
• C. 0
• D. \(\nabla^2 \phi\)

Hint:

The operator \(\nabla^2\) is a scalar operator and is often referred to as the Laplacian. It acts on a scalar function to produce another scalar function. The expression "div grad" is a common way to represent the Laplacian.

Answer 1

The Correct Option is D

Step 1: Understand the operators gradient (\(\nabla\)) and divergence (div or \(\nabla \cdot\)).
- The gradient of a scalar function \(\phi(x, y, z)\) is a vector field defined as: \[ \nabla \phi = \frac{\partial \phi}{\partial x} \mathbf{i} + \frac{\partial \phi}{\partial y} \mathbf{j} + \frac{\partial \phi}{\partial z} \mathbf{k} \] - The divergence of a vector field \(\mathbf{F} = F_x \mathbf{i} + F_y \mathbf{j} + F_z \mathbf{k}\) is a scalar function defined as: \[ \text{div} \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \] Step 2: Apply the divergence operator to the gradient of \(\phi\).
We need to find \(\text{div}(\nabla \phi)\), which is \(\nabla \cdot (\nabla \phi)\). First, let's write out the components of \(\nabla \phi\): \[ F_x = \frac{\partial \phi}{\partial x}, \quad F_y = \frac{\partial \phi}{\partial y}, \quad F_z = \frac{\partial \phi}{\partial z} \] Now, apply the divergence operator to this vector field: \[ \nabla \cdot (\nabla \phi) = \frac{\partial}{\partial x} \left( \frac{\partial \phi}{\partial x} \right) + \frac{\partial}{\partial y} \left( \frac{\partial \phi}{\partial y} \right) + \frac{\partial}{\partial z} \left( \frac{\partial \phi}{\partial z} \right) \] Step 3: Simplify the expression.
The expression obtained in Step 2 can be written using second-order partial derivatives: \[ \nabla \cdot (\nabla \phi) = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} \] Step 4: Recognize the Laplacian operator.
The expression \(\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}\) is the definition of the Laplacian of the scalar function \(\phi\), which is denoted by \(\nabla^2 \phi\) or \(\Delta \phi\). Therefore, \[ \text{div grad } \phi = \nabla \cdot (\nabla \phi) = \nabla^2 \phi \] Step 5: Match the result with the given options.
The result \(\nabla^2 \phi\) matches option (4).