Question

Which of the following statement(s) is/are CORRECT?

• A. Gradient of temperature is a vector.
• B. Gradient of pressure is a vector.
• C. Divergence of velocity is a vector.
• D. Gradient of velocity is a scalar.

Hint:

Vector calculus rules to memorize: - Gradient of scalar → vector. - Divergence of vector → scalar. - Curl of vector → vector. - Gradient of vector → tensor.

Answer 1

The Correct Option is A, B

Step 1: Gradient of a scalar field.
- Temperature \(T(x,y,z)\) and pressure \(P(x,y,z)\) are scalar fields. - The gradient operator is defined as \[ \nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right). \] Thus, \(\nabla T\) and \(\nabla P\) produce vectors pointing in the direction of maximum rate of change. Therefore: - Statement (A) is correct. - Statement (B) is correct. Step 2: Divergence of a vector field.
Velocity \(\vec{v} = (u,v,w)\) is a vector field. The divergence is defined as: \[ \nabla \cdot \vec{v} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}. \] This is a scalar (net volumetric expansion per unit volume). So, statement (C) is incorrect. Step 3: Gradient of a vector field.
The gradient of a vector field is a second-order tensor (matrix of partial derivatives). For velocity, \[ \nabla \vec{v} = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z}
\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z}
\frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z} \end{bmatrix}. \] This is not a scalar. So, statement (D) is incorrect. Final Answer: \[ \boxed{\text{(A) and (B)}} \]