Question

How do you determine if a point \(\left(x_{0}, y_{0}, z_{0}\right)\) in a vector field is a source, a sink, or incompressible?

Short answer

To determine if a point in a vector field is a source, a sink, or incompressible, calculate the divergence of the vector field at that point. The divergence being positive implies the point is a source, negative implies it's a sink and zero reveals the point as incompressible.

Step-by-step solution

Understand Divergence

Divergence of a vector field is a scalar quantity that measures the rate at which 'change' propagates from a given point. It is given by the scalar product of the del operator and the vector field \(F\), denoted as \(\nabla \cdot F\). In Cartesian coordinates \(x, y, z\), the divergence is given by: \(\nabla \cdot F = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\) where \(F_x, F_y, F_z\) are the components of the vector field \(F\).

Compute the Divergence of the Vector Field at the Given Point

Substitute the given point \((x_{0}, y_{0}, z_{0})\) into the divergence expression \(\nabla \cdot F\). Calculate the values for \(\frac{\partial F_x}{\partial x}, \frac{\partial F_y}{\partial y}, \frac{\partial F_z}{\partial z}\) at this specific point.

Interpret the Result

If the divergence is positive at the point, the point is a source because fluid is exiting at a higher rate than entering. If the divergence is negative, the point is a sink because fluid is entering at a higher rate than exiting. If the divergence is zero, fluid is neither created or destroyed at this point - making the point incompressible.