Consider a velocity field u⃗ = (−x + y)ˆex + (−y − x)ˆey
(a) (2 points) Produce a proper quiver plot of u⃗ with a meaningful caption. This velocity field is the superposition of two “simple” fields. Could you guess what they are?

Here is the quiver plot of the velocity field \( \vec{u} = (-x + y)\hat{e_x} + (-y - x)\hat{e_y} \).

### Caption:
The plot represents the velocity field as a combination of two components: a radial inward flow (towards the origin) and a counter-clockwise rotational (vortex-like) flow. The field vectors rotate around the origin while also converging towards it.

Would you like any further details or clarifications?

### Related Questions:
1. How would the velocity field change if the components \( u_x \) and \( u_y \) were swapped?
2. Can you derive the divergence and curl of this velocity field?
3. What is the physical interpretation of this velocity field in a fluid dynamics context?
4. How would the quiver plot change for a field with only a rotational component?
5. Could you find the streamlines of this velocity field?

### Tip:
In fluid mechanics, the superposition of radial and rotational fields often describes more complex flows like vortex flows or saddle points, useful for visualizing fluid behaviors near singularities.

Consider a velocity field u⃗ = (−x + y)ˆex + (−y − x)ˆey. (a) Produce a proper quiver plot of u⃗ with a meaningful caption. This velocity field is the superposition of two 'simple' fields. Could you guess what they are?

Explore the quiver plot of the vector field u⃗ = (−x + y)ˆex + (−y − x)ˆey, which is a combination of radial inward and counter-clockwise rotational flows. This problem covers key concepts in vector calculus and fluid dynamics, with a step-by-step solution. Suitable for undergraduate students.

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