Curl and Divergence — Question 2

Question 2

Consider the position field $$\mathbf F(x,y,z)=\langle x,y,z\rangle.$$

Tasks

1. Compute its divergence and curl.

2. Interpret the divergence as local expansion or contraction.

3. Explain why the field has no local rotational tendency.

Question 2 – Solution

Strategy. Differentiate the matching coordinate in each divergence term and compare the cross derivatives for curl.

Step 1: Divergence $$\nabla\cdot\mathbf F =\frac{\partial x}{\partial x} +\frac{\partial y}{\partial y} +\frac{\partial z}{\partial z} =1+1+1=\boxed{3}.$$

The positive constant divergence describes uniform local expansion: arrows point outward and grow with distance from the origin.

Step 2: Curl Every cross derivative is zero, so $$\nabla\times\mathbf F =\langle 0-0,0-0,0-0\rangle =\boxed{\mathbf 0}.$$

Step 3: Interpret Curl measures infinitesimal rotational tendency. This radial field expands directly away from the origin without circulating around any axis, consistent with zero curl.

Verification The field is $\nabla[(x^2+y^2+z^2)/2]$. A gradient field has zero curl where the second partial derivatives are continuous, confirming the calculation.