Question

The divergence of the vector \(\overrightarrow v = {y^2}\widehat i + {z^2}\widehat j + {x^2}\widehat k\)
1. 2x
2. 2y
3. 2z
4. 0

Answer 1

Correct Answer - Option 4 : 0

Concept:

Divergence:

The divergence of a vector field simply measures how much the flow is expanding at a given point. It does not indicate in which direction the expansion is occurring. Hence (in contrast to the curl of a vector field), the divergence of the vector is a scalar quantity.

In Rectangular coordinates, the divergence is defined as:

 

\(\nabla \cdot \vec v = \left( {\frac{{\partial {v_x}}}{{\partial x}} + \frac{{\partial {v_y}}}{{\partial y}} + \frac{{\partial {v_z}}}{{\partial z}}} \right)\)

Calculation:

Given:

\(\overrightarrow v = {y^2}\widehat i + {z^2}\widehat j + {x^2}\widehat k\)

v_x = y², v_y = z², v_z = x².

Then,

\(\nabla \cdot \vec v = \left( {\frac{{\partial {v_x}}}{{\partial x}} + \frac{{\partial {v_y}}}{{\partial y}} + \frac{{\partial {v_z}}}{{\partial z}}} \right)\)

\(\nabla \cdot \vec v = \left( {\frac{{\partial {y^2}}}{{\partial x}} + \frac{{\partial {z^2}}}{{\partial y}} + \frac{{\partial {x^2}}}{{\partial z}}} \right)\)

= 0.

• Divergence operates on a vector field but results in a scalar.
• Curl operates on a vector field and results in a vector field.
• Gradient operates on a scalar but results in a vector field.
• Divergence of curl, Curl of the gradient is always zero.