Question

Let (, ) be a scalar, vector, or tensor. show that the material derivative is:

A) D(,)/ Dt = ∂(,) / ∂t
B) D(,)/ Dt = d(,) / dt​
C) D(,) / Dt = ∇(,)
D) D(,) / Dt = ∇⋅(,)

Answer 1

The limit lim (x → 0) x²y²ln(x² + y²) does not exist.

Explanation:

To find the limit lim (x → 0) x²y²ln(x² + y²) using polar coordinates, we express x and y in terms of polar coordinates. Let x = r cos(θ) and y = r sin(θ), where r is the distance from the origin and θ is the angle with the positive x-axis.

Substituting these expressions into the limit, we get lim (r → 0) (r cos(θ))² (r sin(θ))² ln((r cos(θ))² + (r sin(θ))²). Simplifying further, we have lim (r → 0) r⁴ ln(r²).

As r approaches 0, r⁴ goes to 0 faster than ln(r²) approaches negative infinity. The product of these two functions leads to an indeterminate form (0 × (-∞)). The limit does not exist because the rate at which r⁴ approaches 0 overwhelms the logarithmic term, resulting in an undefined behavior. Therefore, the limit lim (x → 0) x²y²ln(x² + y²) does not exist.

Answer 2

The correct expression for the material derivative is given by option A) D(,) / Dt = ∂(,) / ∂t.

Step-by-step explanation:

The material derivative, denoted as D(,)/Dt, represents the rate of change of a property with respect to time as observed by an observer moving with the material. The material derivative includes both the local time rate of change (partial derivative with respect to time, ∂(,)/∂t) and the convective or advective contribution due to the motion of the material. In the given options, option A accurately represents the material derivative, indicating that it consists of the partial derivative with respect to time.

To illustrate, consider a scalar, vector, or tensor field denoted by φ. The material derivative of φ, D(φ)/Dt, is given by the sum of the partial derivative of φ with respect to time (∂φ/∂t) and the convective term involving the velocity field, typically denoted as V. Mathematically, D(φ)/Dt = ∂φ/∂t + V ⋅ ∇φ, where ∇φ is the gradient of φ. In the case of a scalar field, this expression reduces to D(φ)/Dt = ∂φ/∂t. This supports the correctness of option A.

Understanding the material derivative is crucial in fluid dynamics, continuum mechanics, and various branches of physics where the motion and deformation of materials are analyzed. The material derivative accounts for both local changes and changes due to the motion of the material, providing a comprehensive measure of how properties evolve over time.