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:

In vector calculus, gradients are always vectors, divergence is a scalar, and the gradient of a vector field like velocity is a tensor. Understanding these concepts is essential for fluid dynamics and many other fields in engineering.

Answer 1

The Correct Option is A, B

In vector calculus, gradients, divergences, and curls are all important operations used to describe physical quantities in space, and each has its own specific behavior. 
- The gradient of temperature (\( \nabla T \)) is a vector because it describes the rate of change of temperature with respect to position in a specific direction. The gradient gives both the magnitude and direction of the temperature change, which makes it a vector field. 
- The gradient of pressure (\( \nabla P \)) is also a vector because pressure changes in space and is directional. Just like temperature, the gradient of pressure indicates the direction of the highest rate of pressure change, making it a vector field. 
- The divergence of velocity is a scalar, not a vector. Divergence measures the net flow (expansion or contraction) of a vector field at a point. It is used to quantify the amount of flow in or out of a point, and it results in a scalar value. 
- The gradient of velocity is a tensor, not a scalar. The gradient of the velocity vector represents how the components of the velocity vector change in space and is more complex than just a scalar or a vector. Thus, the correct statements are options (A) and (B), as both gradients of temperature and pressure are vector fields.