MATH1011 - Lecture 20

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MATH1011 - Lecture 19
Skeleton Slides

Flux Across a Curve

• In ${\mathbb{R}}^2$ to calculate circulation, we integrated the tangential component of the vector field
  • Now we are going to integrate the normal component to find flux

Flux vs Circulation

Circulation is a measure of flow over a point (forwards and backwards)

• It is applied in ${\mathbb{R}}^2$

Flux is a measure of flow through a point (inwards and outwards)

• The flux of $\mathbf{F}$ across $\mathbf{C}$ is:$\underset{C}{\int }\mathbf{F}\cdot \mathbf{n}ds={\int }_a^b\mathbf{F}(\mathbf{r}(t))\cdot \mathbf{n}|\dot{\mathbf{r}}(t)|dt$
  • We denote $\mathbf{n}|\dot{\mathbf{r}}(t)|$ as $\mathbf{N}(t)$: it points in the same direction as $\mathbf{n}$ and its norm is the same as the norm of $\dot{\mathbf{r}}(t)$

Flux Across a Curve in ${\mathbb{R}}^2$

• Suppose $\dot{\mathbf{r}}(t)=(c(t),d(t))$, then there are two vectors normal to it:
  • $(-d(t),c(t))$ and $(d(t),-c(t))$
• $\mathbf{N}(t)=(d(t),-c(t))$
  • This simplifies finding $\mathbf{N}$ when working in ${\mathbb{R}}^2$

Similarly to calculating circulation, you can split up a flux integral if the curve $C$ is piecewise

Flux Across a Surface in ${\mathbb{R}}^3$

• We are given tangent vectors ${\mathbf{S}}_u$ and ${\mathbf{S}}_v$ at $\mathbf{S}(u,v)$
  • ${\mathbf{S}}_u=\frac{\partial S}{\partial u}$ and ${\mathbf{S}}_v=\frac{\partial S}{\partial v}$
• $\mathbf{N}={\mathbf{S}}_u\times {\mathbf{S}}_v$ is normal to the surface

How do we find the normal component of $\mathbf{F}$ at $\mathbf{S}(u,v)$?
$\mathbf{N}=\mathbf{F}(\mathbf{S}(u,v))\cdot \mathbf{n}$

• There are two unit vectors $\mathbf{n}$ are normal to the surface:
  $\frac{\mathbf{N}}{|\mathbf{N}|}\text{and}\frac{-\mathbf{N}}{|\mathbf{N}|}$

We need a convention to choose $\mathbf{n}$:

• $S$ is a bounding surface if it is the boundary of a solid region of $\mathbb{R}$

  • If $S$ is a bounding surface then by convention we choose $\mathbf{n}$ pointing outwards

• Upwards means towards positive $z$

• We define the flux as the surface integral:
  $\underset{D}{\iint }\mathbf{F}\cdot \mathbf{N}dudv$

• Note: $\mathbf{F}=f(\mathbf{S}(u,v))$

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MATH1011 - Lecture 21
Annotated Slides