Question

Consider a Cartesian coordinate system with orthogonal unit basis vectors \( \hat{i}, \hat{j} \) defined over a domain: \( x, y \in [0,1] \). Choose the condition for which the divergence of the vector field \( \mathbf{v} = ax\hat{i} - by\hat{j} \) is zero.

• A. \( a - b = 0 \)
• B. \( a<b \)
• C. \( a>b \)
• D. \( a + b = 0 \)

Hint:

The divergence of a vector field is a key concept in vector calculus, measuring the net flow of a vector field through an infinitesimal volume. In this case, the divergence becomes zero when the coefficients of the vector field are equal.

Answer 1

The Correct Option is A

To find the divergence of the vector field \( \mathbf{v} = ax\hat{i} - by\hat{j} \), we calculate the divergence using the definition: \[ \nabla \cdot \mathbf{v} = \frac{\partial (ax)}{\partial x} + \frac{\partial (-by)}{\partial y} \] Evaluating the partial derivatives: \[ \frac{\partial (ax)}{\partial x} = a, \quad \frac{\partial (-by)}{\partial y} = -b \] Thus, the divergence of \( \mathbf{v} \) is: \[ \nabla \cdot \mathbf{v} = a - b \] For the divergence to be zero: \[ a - b = 0 \Rightarrow a = b \] This simplifies to the condition where \( a \) must equal \( b \), verifying option (A) as correct.