Which of the following statement(s) is/are CORRECT?
Show Hint
- Gradient of temperature is a vector.
- Gradient of pressure is a vector.
- Divergence of velocity is a vector.
- Gradient of velocity is a scalar.
The Correct Option is A, B
Solution and Explanation
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.
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