Question

The value of \( v_3 \) for which the vector \( \vec{v} = e^y \sin x \hat{i} + e^y \cos x \hat{j} + v_3 \hat{k} \) is solenoidal, is:

• A. \( 2ze^y \cos x \)
• B. \( -2ze^y \cos x \)
• C. \( -2e^y \cos x \)
• D. \( 2e^y \sin x \)

Hint:

Remember the physical meanings of the vector operators:
\textbf{Divergence (\(\nabla \cdot \vec{v}\))}: Measures the rate of expansion from a point (source or sink). Solenoidal (\(\nabla \cdot \vec{v} = 0\)) means the field is incompressible.
\textbf{Curl (\(\nabla \times \vec{v}\))}: Measures the rotation or circulation at a point. Irrotational (\(\nabla \times \vec{v} = \vec{0}\)) means the field is conservative.

Answer 1

The Correct Option is B

Step 1: Definition:
A solenoidal vector field \( \vec{v} \) has a divergence of zero everywhere. The divergence of \( \vec{v} = v_1 \hat{i} + v_2 \hat{j} + v_3 \hat{k} \) is \( abla \cdot \vec{v} = \frac{\partial v_1}{\partial x} + \frac{\partial v_2}{\partial y} + \frac{\partial v_3}{\partial z} \).

Step 2: Method:
Given \( v_1 = e^y \sin x \), \( v_2 = e^y \cos x \), and an unknown \( v_3 \), find \( v_3 \) such that the divergence of \( \vec{v} \) is zero. \[ abla \cdot \vec{v} = \frac{\partial}{\partial x}(e^y \sin x) + \frac{\partial}{\partial y}(e^y \cos x) + \frac{\partial v_3}{\partial z} = 0 \]
Step 3: Solution:
Calculate the partial derivatives: \[ \frac{\partial v_1}{\partial x} = \frac{\partial}{\partial x}(e^y \sin x) = e^y \cos x \]\[ \frac{\partial v_2}{\partial y} = \frac{\partial}{\partial y}(e^y \cos x) = e^y \cos x \]Substitute into the divergence equation: \[ (e^y \cos x) + (e^y \cos x) + \frac{\partial v_3}{\partial z} = 0 \]\[ 2e^y \cos x + \frac{\partial v_3}{\partial z} = 0 \]Solve for \( v_3 \): \[ \frac{\partial v_3}{\partial z} = -2e^y \cos x \]Integrate with respect to \(z\): \[ v_3(x,y,z) = \int (-2e^y \cos x) dz = -2e^y \cos x \int dz = -2ze^y \cos x + C(x,y) \]Choose the simplest form, \( C(x,y) = 0 \). Thus, \( v_3 = -2ze^y \cos x \).

Step 4: Answer:
\(v_3 = -2ze^y \cos x \)