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GATE 2021 MT – Question 13

Engineering Mathematics · Calculus and vector calculus · 1 mark · Multiple choice

For vector field $\mathbf V(x,y,z)$ with components $V_x,V_y,V_z$ depending on x,y,z, its divergence is ______.

  1. $\partial_xV_x+\partial_yV_y+\partial_zV_z$
  2. $(\partial_yV_z-\partial_zV_y)\mathbf i+(\partial_zV_x-\partial_xV_z)\mathbf j+(\partial_xV_y-\partial_yV_x)\mathbf k$
  3. $(\partial_xV_x)\mathbf i+(\partial_yV_y)\mathbf j+(\partial_zV_z)\mathbf k$
  4. $\partial_x^2V_x+\partial_y^2V_y+\partial_z^2V_z$

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Correct answer: (A) $\partial_xV_x+\partial_yV_y+\partial_zV_z$

Explanation

Divergence sums each component’s derivative along its corresponding coordinate: **∂Vx/∂x+∂Vy/∂y+∂Vz/∂z, A**. Cross derivatives instead enter curl, and second derivatives enter a Laplacian.