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Understanding Vector Fields: Divergence and Curl Explained

This document provides an overview of key concepts in vector fields, focusing on divergence and curl. The divergence of a vector field at a point indicates the net flow of fluid, allowing us to classify points as sources, sinks, or incompressible regions. Additionally, we discuss curl, which measures the rotation of fluid at a point and is determined using the right-hand rule. Examples are provided, including calculations for vector fields F(x, y, z) to illustrate these concepts, enhancing comprehension of fluid dynamics and vector calculus.

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Understanding Vector Fields: Divergence and Curl Explained

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  1. F (x, y) = <y, -x> F (x, y) = <x, y> Div F = 0 Div F = 2 > 0 The divergence of a vector field at a point (x, y, z) corresponds to the net flow Of fluid out of a small box centered at (x, y, z). If Div F > 0, more fluid exits the Box than enters (as in the figure on the left) and we call the point a source. If Div F <0, more fluid enters the box than exits and we call the point a sink. If Div F = 0, throughout some region D, then we say that the vector field F is Source-free or incompressible.

  2. F (x, y, z) = <x, y, 0> Curl F = <0, 0, 0> F (x, y, z) = <y, -x, 0> Curl F = <0, 0, -2> The curl of a vector filed at a point measures the rotation of the fluid at the point. The direction of the vector Curl F is determined by the right hand rule (curl the Fingers of your right hand along the flow lines with your fingertips pointing in the Direction of the flow, your thumb will point to the direction that is parallel to Curl F. If Curl F = zero vector, then the vector field F is irrotational (as in the figure on the Left).

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