Divergence of a vector field evaluated at a point
BlinkNBuild practice problem · GATE standard, authored and verified in-house
For the vector field $$\vec{F} = x^2 y\,\hat{i} + y z\,\hat{j} + x z^2\,\hat{k},$$ the value of $\nabla \cdot \vec{F}$ at the point $(1,\,1,\,1)$ is ________.
Show the step-by-step derivation
Answer
5 unitless unitless
Step-by-step derivation
- Divergence is the sum of the partial derivative of each component with respect to its own coordinate: $$\nabla\cdot\vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}.$$
- First term, treating $y$ as constant: $$\frac{\partial}{\partial x}\left(x^2 y\right) = 2xy.$$
- Second term, treating $z$ as constant: $$\frac{\partial}{\partial y}\left(y z\right) = z.$$
- Third term, treating $x$ as constant: $$\frac{\partial}{\partial z}\left(x z^2\right) = 2xz.$$
- Assemble the divergence field: $$\nabla\cdot\vec{F} = 2xy + z + 2xz.$$
- Evaluate at $(1,1,1)$: $$\nabla\cdot\vec{F}\big|_{(1,1,1)} = 2(1)(1) + 1 + 2(1)(1) = 2 + 1 + 2 = \mathbf{5}.$$
- Sanity check on the result type. Divergence maps a vector field to a scalar field, so a single number is the right kind of answer. Curl would have produced a vector - if you find yourself with three components, you computed the wrong operator.
The trap this question is built around
Differentiating each component with respect to the wrong variable - for example taking $\partial(yz)/\partial z$ instead of $\partial(yz)/\partial y$. Divergence pairs $F_x$ with $x$, $F_y$ with $y$ and $F_z$ with $z$; any other pairing belongs to the curl.