MATH 251 Lecture 30

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Divergence

Start with a vector field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F}(x,y) = a(x,y) \hat\imath + b(x,y) \hat\jmath}

Divergence represents the net flow out of a point (x,y). This is a scalar field, which is a regular old function.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{div} \vec{F} = \nabla \cdot \vec{F} = \frac{\partial a}{\partial x} + \frac{\partial b}{\partial y}}

Think of the vector field as the flow of a fluid.

A source point is a point at which there is more fluid moving away than moving toward: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \vec{F} > 0}

A sink point is a point at which there is more fluid moving toward than away: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \vec{F} < 0}

An incompressible point is a point at which there is an equal amount of fluid moving toward and away: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \vec{F} = 0}

Derivation

Measure fluid produced in a small box around point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0,y_0)} . The box is a square from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0-h, y_0-h)} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0+h, y_0+h)} .

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\hat\imath} represent the difference between fluid out and fluid in (out − in) in the x-direction.

  • Right: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(x_0+h, y_0)\cdot(2h)}
  • Top: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b(x_0, y_0+h)\cdot(2h)}
  • Left: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -a(x_0-h, y_0)\cdot(2h)}
  • Bottom: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -b(x_0, y_0-h) \cdot (2h)}

The sum of all of these equals the area of the box (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4h^2} ) times average production in box (APB). Solve for average production to get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{APB} = \frac{a(x_0+h,y_0) - a(x_0-h,y_0)}{2h} + \frac{b(x_0,y_0+h) - b(x_0,y_0-h)}{2h}}

Take the limit as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \to 0} , and this turns into a derivative:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial a}{\partial x} + \frac{\partial b}{\partial y}}

Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \vec{F} &= 3x^2 \hat\imath - (x+4y) \hat\imath \\ \nabla \cdot \vec{F} &= 6x - 4 \end{align}}


Explanation of Notation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla = \frac{\partial}{\partial x} \hat\imath + \frac{\partial}{\partial y} \hat\jmath} is a "differential operator":

gradient
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla f = \frac{\partial f}{\partial x} \hat\imath + \frac{\partial f}{\partial y} \hat\jmath}
divergence
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \nabla \cdot \vec{F} &= \left( \frac{\partial}{\partial x} \hat\imath + \frac{\partial}{\partial y} \hat\jmath \right) \cdot \left( a \hat\imath + b \hat\jmath \right) \\ &= \frac{\partial a}{\partial x} + \frac{\partial b}{\partial y} \end{align}}


Curl

Measures the circulation (imagine a vortex) of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F}} around a point (by the right hand rule). In 3D space, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \nabla \times \vec{F} \right) \cdot \hat{n}} gives the circulation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F}} around Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{n}} axis.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial_x = \tfrac{\partial}{\partial x}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \nabla \times \vec{F} &= \left( \partial_x \hat\imath + \partial_y \hat\jmath + \partial_k \hat{k} \right) \times \left( a \hat\imath + b \hat\jmath + c \hat{k} \right) \\ &= \begin{vmatrix} \hat\imath & \hat\jmath & \hat{k} \\ \partial_x & \partial_y & \partial_z \\ a & b & c \end{vmatrix} \end{align}}

Special Case

When Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F} = a(x,y) \hat\imath + b(x,y) \hat\jmath} , the curl is equal to:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{vmatrix} \hat\imath & \hat\jmath & \hat{k} \\ \partial_x & \partial_y & \partial_z \\ a & b & 0 \end{vmatrix} = \left( \frac{\partial b}{\partial x} - \frac{\partial x}{\partial y} \right) \hat{k}}

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