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quarto/integral_vector_calculus/div_grad_curl.qmd
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# The Gradient, Divergence, and Curl
# The gradient, divergence, and curl
{{< include ../_common_code.qmd >}}

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quarto/integral_vector_calculus/line_integrals.qmd
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# Line and Surface Integrals
# Line and surface integrals
{{< include ../_common_code.qmd >}}
@@ -340,7 +340,7 @@ W = integrate(F(r(t)) ⋅ T(r(t)), (t, 0, 2PI))
There are technical assumptions about curves and regions that are necessary for some statements to be made:
* Let $C$ be a [Jordan](https://en.wikipedia.org/wiki/Jordan_curve_theorem) curve - a non-self-intersecting continuous loop in the plane. Such a curve divides the plane into two regions, one bounded and one unbounded. The normal to a Jordan curve is assumed to be in the direction of the unbounded part.
* Let $C$ be a [Jordan](https://en.wikipedia.org/wiki/Jordan_curve_theorem) curve---a non-self-intersecting continuous loop in the plane. Such a curve divides the plane into two regions, one bounded and one unbounded. The normal to a Jordan curve is assumed to be in the direction of the unbounded part.
* Further, we will assume that our curves are *piecewise smooth*. That is comprised of finitely many smooth pieces, continuously connected.
* The region enclosed by a closed curve has an *interior*, $D$, which we assume is an *open* set (one for which every point in $D$ has some "ball" about it entirely within $D$ as well.)
* The region $D$ is *connected* meaning between any two points there is a continuous path in $D$ between the two points.
@@ -471,7 +471,7 @@ The flow integral is typically computed for a closed (Jordan) curve, measuring t
:::{.callout-note}
## Note
For a Jordan curve, the positive orientation of the curve is such that the normal direction (proportional to $\hat{T}'$) points away from the bounded interior. For a non-closed path, the choice of parameterization will determine the normal and the integral for flow across a curve is dependent - up to its sign - on this choice.
For a Jordan curve, the positive orientation of the curve is such that the normal direction (proportional to $\hat{T}'$) points away from the bounded interior. For a non-closed path, the choice of parameterization will determine the normal and the integral for flow across a curve is dependent---up to its sign---on this choice.
:::

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quarto/integral_vector_calculus/review.qmd
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# Quick Review of Vector Calculus
# Quick review of vector calculus
{{< include ../_common_code.qmd >}}
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$$
The generalization to $n>2$ is clear - the partial derivative in $x_i$ is the derivative of $f$ when the *other* $x_j$ are held constant.
The generalization to $n>2$ is clear---the partial derivative in $x_i$ is the derivative of $f$ when the *other* $x_j$ are held constant.
This may be viewed as the derivative of the univariate function $(f\circ\vec{r})(t)$ where $\vec{r}(t) = p + t \hat{e}_i$, $\hat{e}_i$ being the unit vector of all $0$s except a $1$ in the $i$th component.

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quarto/integral_vector_calculus/stokes_theorem.qmd
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# Green's Theorem, Stokes' Theorem, and the Divergence Theorem
# Green's theorem, Stokes' theorem, and the divergence theorem
{{< include ../_common_code.qmd >}}
@@ -721,13 +721,13 @@ The fluid would flow along the blue (stream) lines. The red lines have equal pot
# https://en.wikipedia.org/wiki/Jiffy_Pop#/media/File:JiffyPop.jpg
imgfile ="figures/jiffy-pop.png"
caption ="""
The Jiffy Pop popcorn design has a top surface that is designed to expand to accommodate the popped popcorn. Viewed as a surface, the surface area grows, but the boundary - where the surface meets the pan - stays the same. This is an example that many different surfaces can have the same bounding curve. Stokes' theorem will relate a surface integral over the surface to a line integral about the bounding curve.
The Jiffy Pop popcorn design has a top surface that is designed to expand to accommodate the popped popcorn. Viewed as a surface, the surface area grows, but the boundary---where the surface meets the pan---stays the same. This is an example that many different surfaces can have the same bounding curve. Stokes' theorem will relate a surface integral over the surface to a line integral about the bounding curve.
"""
# ImageFile(:integral_vector_calculus, imgfile, caption)
nothing
```
![The Jiffy Pop popcorn design has a top surface that is designed to expand to accommodate the popped popcorn. Viewed as a surface, the surface area grows, but the boundary - where the surface meets the pan - stays the same. This is an example that many different surfaces can have the same bounding curve. Stokes' theorem will relate a surface integral over the surface to a line integral about the bounding curve.
![The Jiffy Pop popcorn design has a top surface that is designed to expand to accommodate the popped popcorn. Viewed as a surface, the surface area grows, but the boundary---where the surface meets the pan---stays the same. This is an example that many different surfaces can have the same bounding curve. Stokes' theorem will relate a surface integral over the surface to a line integral about the bounding curve.
](./figures/jiffy-pop.png)
Were the figure of Jiffy Pop popcorn animated, the surface of foil would slowly expand due to pressure of popping popcorn until the popcorn was ready. However, the boundary would remain the same. Many different surfaces can have the same boundary. Take for instance the upper half unit sphere in $R^3$ it having the curve $x^2 + y^2 = 1$ as a boundary curve. This is the same curve as the surface of the cone $z = 1 - (x^2 + y^2)$ that lies above the $x-y$ plane. This would also be the same curve as the surface formed by a Mickey Mouse glove if the collar were scaled and positioned onto the unit circle.
@@ -761,7 +761,7 @@ $$
$$
In terms of our expanding popcorn, the boundary integral - after accounting for cancellations, as in Green's theorem - can be seen as a microscopic sum of boundary integrals each of which is approximated by a term $\nabla\times{F}\cdot\hat{N} \Delta{S}$ which is viewed as a Riemann sum approximation for the the integral of the curl over the surface. The cancellation depends on a proper choice of orientation, but with that we have:
In terms of our expanding popcorn, the boundary integral---after accounting for cancellations, as in Green's theorem---can be seen as a microscopic sum of boundary integrals each of which is approximated by a term $\nabla\times{F}\cdot\hat{N} \Delta{S}$ which is viewed as a Riemann sum approximation for the the integral of the curl over the surface. The cancellation depends on a proper choice of orientation, but with that we have:
::: {.callout-note icon=false}
## Stokes' theorem