New derivatives and the del operator

Section 17.3 New derivatives and the del operator

Objectives

Define divergence and curl as measures of outward flow and rotational behavior.
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Interpret incompressible and irrotational vector fields as ideal fluid flows.
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Use the del operator to express gradient, divergence, and curl in unified notation.
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Introduction goes here.

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Divergence of a vector field:
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\begin{align*} \Div(P,Q) \amp = \pdv{P}{x}+\pdv{Q}{y}\\ \Div(P,Q,R) \amp = \pdv{P}{x}+\pdv{Q}{y}+\pdv{R}{z} \end{align*}

This measures how much a vector field is ‘flowing outward’. If the divergence in a region is positive, it’s acting like a source, while if it’s negative, it’s acting like a sink.
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Curl of a vector field:
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\begin{align*} \Curl(P,Q) \amp = \pdv{Q}{x}-\pdv{P}{y}\\ \Curl(P,Q,R) \amp = \qty(\pdv{R}{y}-\pdv{Q}{z},\pdv{P}{z}-\pdv{R}{x},\pdv{Q}{x}-\pdv{P}{y}) \end{align*}

This measures how much a vector field is ‘rotating’.
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A vector field with zero divergence is called incompressible or solenoidal.
- In the 2D case, the curl is a scalar. If curl is positive, the field is rotating counterclockwise, while if negative, it’s rotating clockwise.
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- In the 3D case, the curl is a vector, which points along the direction of the right-hand rule, so that the field is flowing counterclockwise around that vector as an axis. The magnitude is the strength of rotation.
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A vector field with zero curl is called irrotational.

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In fluid dynamics, an ideal fluid is incompressible and irrotational.
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Some useful properties:

\begin{equation*} \Curl(\Grad f)=\bv 0\qquad\Div(\Curl\bv F)=0 \end{equation*}

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The del operator \(\nabla\) is a vector operator.
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In 2D:

\begin{equation*} \nabla=\qty(\pdv{x},\pdv{y}) \end{equation*}

In 3D:

\begin{equation*} \nabla=\qty(\pdv{x},\pdv{y},\pdv{z}) \end{equation*}

With this operator, we have:

\begin{align*} \Grad f \amp = \nabla f\\ \Div\bv{F} \amp = \nabla\cdot\bv{F}\\ \Curl\bv{F} \amp = \nabla\times\bv{F} \end{align*}

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Divergence and curl act like derivatives in that they distribute over sums and satisfy a product rule in many cases:
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\begin{align*} \Div(\bv{F}+\bv{G}) \amp = \Div\bv{F}+\Div\bv{G}\\ \Curl(\bv{F}+\bv{G}) \amp = \Curl\bv{F}+\Curl\bv{G}\\ \Div(f\bv{F}) \amp = f\Div\bv{F}+\nabla f\cdot \bv{F}\\ \Curl(f\bv{F}) \amp = f\Curl\bv{F}+\nabla f\times\bv{F} \end{align*}

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https://mathintuitions.com/2025/05/27/divergence-curl-and-the-taylor-series-approximation/
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Include polar versions as an exercise
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