CalculusWithJuliaNotes.jl/CwJ/integral_vector_calculus/review.jmd

# Quick Review of Vector Calculus


```julia; echo=false; results="hidden"
using CalculusWithJulia
using CalculusWithJulia.WeaveSupport
using Plots

const frontmatter = (
        title = "Quick Review of Vector Calculus",
        description = "Calculus with Julia: Quick Review of Vector Calculus",
        tags = ["CalculusWithJulia", "integral_vector_calculus", "quick review of vector calculus"],
);

nothing
```


This section considers functions from ``R^n`` into ``R^m`` where one or both of ``n`` or ``m`` is greater than ``1``:

* functions ``f:R \rightarrow R^m`` are called univariate functions.

* functions ``f:R^n \rightarrow R`` are called scalar-valued functions.

* function ``f:R \rightarrow R`` are univariate, scalar-valued functions.

* functions ``\vec{r}:R\rightarrow R^m`` are parameterized curves. The trace of a parameterized curve is a path.

* functions ``F:R^n \rightarrow R^m``, may be called vector fields in applications. They are also used to describe transformations.

When ``m>1`` a function is called *vector valued*.

When ``n>1`` the argument may be given in terms of components, e.g. ``f(x,y,z)``; with a point as an argument, ``F(p)``; or with a vector as an argument, ``F(\vec{a})``. The identification of a point with a vector is done frequently.

## Limits

Limits when ``m > 1`` depend on the limits of each component existing.

Limits when ``n > 1`` are more complicated. One characterization is a limit at a point ``c`` exists if and only if for *every* continuous path going to ``c`` the limit along the path for every component exists in the univariate sense.


## Derivatives

The derivative of a univariate function, ``f``, at a point ``c`` is defined by a limit:

```math
f'(c) = \lim_{h\rightarrow 0} \frac{f(c+h)-f(c)}{h},
```

and as a function by considering the mapping ``c`` into ``f'(c)``. A characterization is it is the value for which

```math
|f(c+h) - f(h) - f'(c)h| = \mathcal{o}(|h|),
```

That is,  after dividing the left-hand side by ``|h|`` the expression goes to ``0`` as ``|h|\rightarrow 0``. This characterization will generalize with the norm replacing the absolute value, as needed.


### Parameterized curves

The derivative of a function ``\vec{r}: R \rightarrow R^m``, ``\vec{r}'(t)``, is found by taking the derivative of each component. (The function consisting of just one component is univariate.)

The derivative satisfies

```math
\| \vec{r}(t+h) - \vec{r}(t) - \vec{r}'(t) h \| = \mathcal{o}(|h|).
```

The derivative is *tangent* to the curve and indicates the direction of travel.

The **tangent** vector is the unit vector in the direction of ``\vec{r}'(t)``:

```math
\hat{T} = \frac{\vec{r}'(t)}{\|\vec{r}(t)\|}.
```

The path is parameterized by *arc* length if ``\|\vec{r}'(t)\| = 1`` for all ``t``. In this case an "``s``" is used for the parameter, as a notational hint: ``\hat{T} = d\vec{r}/ds``.

The **normal** vector is the unit vector in the direction of the derivative of the tangent vector:

```math
\hat{N} = \frac{\hat{T}'(t)}{\|\hat{T}'(t)\|}.
```

In dimension ``m=2``, if ``\hat{T} = \langle a, b\rangle`` then ``\hat{N} = \langle -b, a\rangle`` or ``\langle b, -a\rangle`` and ``\hat{N}'(t)`` is parallel to ``\hat{T}``.

In dimension ``m=3``, the **binormal** vector, ``\hat{B}``, is the unit vector ``\hat{T}\times\hat{N}``.

The [Frenet-Serret]() formulas define the **curvature**, ``\kappa``,  and the **torsion**, ``\tau``, by

```math
\begin{align}
\frac{d\hat{T}}{ds} &=  & \kappa \hat{N} &\\
\frac{d\hat{N}}{ds} &= -\kappa\hat{T} & & + \tau\hat{B}\\
\frac{d\hat{B}}{ds} &= & -\tau\hat{N}&
\end{align}
```

These formulas apply in dimension ``m=2`` with ``\hat{B}=\vec{0}``.

The curvature, ``\kappa``, can be visualized by imagining a circle of radius ``r=1/\kappa`` best approximating the path at a point. (A straight line would have a circle of infinite radius and curvature ``0``.)


The chain rule says ``(\vec{r}(g(t))' = \vec{r}'(g(t)) g'(t)``.


### Scalar functions

A scalar function, ``f:R^n\rightarrow R``, ``n > 1`` has a **partial derivative** defined. For ``n=2``, these are:

```math
\begin{align}
\frac{\partial{f}}{\partial{x}}(x,y) &=
\lim_{h\rightarrow 0} \frac{f(x+h,y)-f(x,y)}{h}\\
\frac{\partial{f}}{\partial{y}}(x,y) &=
\lim_{h\rightarrow 0} \frac{f(x,y+h)-f(x,y)}{h}.
\end{align}
```

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.


The **gradient** of ``f``, when the limits exist, is the vector-valued function for ``R^n`` to ``R^n``:

```math
\nabla{f} = \langle
\frac{\partial{f}}{\partial{x_1}},
\frac{\partial{f}}{\partial{x_2}},
\dots
\frac{\partial{f}}{\partial{x_n}}
\rangle.
```

The gradient satisfies:

```math
\|f(\vec{x}+\Delta{\vec{x}}) - f(\vec{x}) - \nabla{f}\cdot\Delta{\vec{x}}\| = \mathcal{o}(\|\Delta{\vec{x}\|}).
```

The gradient is viewed as a column vector. If the dot product above is viewed as matrix multiplication, then it would be written ``\nabla{f}' \Delta{\vec{x}}``.


**Linearization** is the *approximation*

```math
f(\vec{x}+\Delta{\vec{x}}) \approx f(\vec{x}) + \nabla{f}\cdot\Delta{\vec{x}}.
```

The **directional derivative** of ``f`` in the direction ``\vec{v}`` is ``\vec{v}\cdot\nabla{f}``, which can be seen as the derivative of the univariate function ``(f\circ\vec{r})(t)`` where ``\vec{r}(t) = p + t \vec{v}``.


For the function ``z=f(x,y)`` the gradient points in the direction of steepest ascent. Ascent is seen in the ``3``d surface, the gradient is ``2`` dimensional.

For a function ``f(\vec{x})``, a **level curve** is the set of values for which ``f(\vec{x})=c``, ``c`` being some constant. Plotted, this may give a curve or surface (in ``n=2`` or ``n=3``). The gradient at a point ``\vec{x}`` with ``f(\vec{x})=c`` will be *orthogonal* to the level curve ``f=c``.


Partial derivatives are scalar functions, so will themselves have partial derivatives when the limits are defined. The notation ``f_{xy}`` stands for the partial derivative in ``y`` of the partial derivative of ``f`` in ``x``. [Schwarz]()'s theorem says the order of partial derivatives will not matter (e.g., ``f_{xy} = f_{yx}``) provided the higher-order derivatives are continuous.


The chain rule applied to ``(f\circ\vec{r})(t)`` says:

```math
\frac{d(f\circ\vec{r})}{dt} = \nabla{f}(\vec{r}) \cdot \vec{r}'.
```


### Vector-valued functions

For a function ``F:R^n \rightarrow R^m``, the **total derivative** of ``F`` is the linear operator ``d_F`` satisfying:

```math
\|F(\vec{x} + \vec{h})-F(\vec{x}) - d_F \vec{h}\| = \mathcal{o}(\|\vec{h}\|)
```


For ``F=\langle f_1, f_2, \dots, f_m\rangle`` the total derivative is the  **Jacobian**, a ``m \times n`` matrix of partial derivatives:

```math
J_f = \left[
\begin{align}{}
\frac{\partial f_1}{\partial x_1} &\quad \frac{\partial f_1}{\partial x_2} &\dots&\quad\frac{\partial f_1}{\partial x_n}\\
\frac{\partial f_2}{\partial x_1} &\quad \frac{\partial f_2}{\partial x_2} &\dots&\quad\frac{\partial f_2}{\partial x_n}\\
&&\vdots&\\
\frac{\partial f_m}{\partial x_1} &\quad \frac{\partial f_m}{\partial x_2} &\dots&\quad\frac{\partial f_m}{\partial x_n}
\end{align}
\right].
```

This can be viewed as being comprised of row vectors, each being the individual gradients; or as column vectors each being the vector of partial derivatives for a given variable.


The **chain rule** for ``F:R^n \rightarrow R^m`` composed with ``G:R^k \rightarrow R^n`` is:

```math
d_{F\circ G}(a) = d_F(G(a)) d_G(a),
```

That is the total derivative of ``F`` at the point ``G(a)`` times (matrix multiplication) the total derivative of ``G`` at ``a``.  The dimensions work out as ``d_F`` is ``m\times n`` and ``d_G`` is ``n\times k``, so ``d_(F\circ G)`` will be ``m\times k`` and ``F\circ{G}: R^k\rightarrow R^m``.

A scalar function ``f:R^n \rightarrow R`` and a parameterized curve ``\vec{r}:R\rightarrow R^n`` composes to yield a univariate function. The total derivative of ``f\circ\vec{r}`` satisfies:

```math
d_f(\vec{r}) d_\vec{r} = \nabla{f}(\vec{r}(t))' \vec{r}'(t) =
\nabla{f}(\vec{r}(t)) \cdot \vec{r}'(t),
```
as above. (There is an identification of a ``1\times 1`` matrix with a scalar in re-expressing as a dot product.)

### The divergence, curl, and their vanishing properties

Define the **divergence** of a vector-valued function ``F:R^n \rightarrow R^n`` by:

```math
\text{divergence}(F) =
\frac{\partial{F_{x_1}}}{\partial{x_1}} +
\frac{\partial{F_{x_2}}}{\partial{x_2}} + \cdots
\frac{\partial{F_{x_n}}}{\partial{x_n}}.
```

The divergence is a scalar function. For a vector field ``F``, it measures the microscopic flow out of a region.

A vector field whose divergence is identically ``0`` is called **incompressible**.

Define the **curl** of a *two*-dimensional vector field, ``F:R^2 \rightarrow R^2``, by:

```math
\text{curl}(F) = \frac{\partial{F_y}}{\partial{x}} -
\frac{\partial{F_x}}{\partial{y}}.
```

The curl for ``n=2`` is a scalar function.

For ``n=3`` define the **curl** of ``F:R^3 \rightarrow R^3`` to be the *vector field*:

```math
\text{curl}(F) =
\langle \
\frac{\partial{F_z}}{\partial{y}} - \frac{\partial{F_y}}{\partial{z}},
\frac{\partial{F_x}}{\partial{z}} - \frac{\partial{F_z}}{\partial{x}},
\frac{\partial{F_y}}{\partial{x}} - \frac{\partial{F_x}}{\partial{y}}
\rangle.
```

The curl measures the circulation in a vector field. In dimension ``n=3`` it *points* in the direction of the normal of the plane of maximum circulation with direction given by the right-hand rule.


A vector field whose curl is identically of magnitude ``0`` is called **irrotational**.

The ``\nabla`` operator is the *formal* vector
```math
\nabla = \langle
\frac{\partial}{\partial{x}},
\frac{\partial}{\partial{y}},
\frac{\partial}{\partial{z}}
\rangle.
```

The gradient is then scalar "multiplication" on the left: ``\nabla{f}``.

The divergence is the dot product on the left: ``\nabla\cdot{F}``.

The curl is the the cross product on the left: ``\nabla\times{F}``.


These operations satisfy two vanishing properties:

* The curl of a gradient is the zero vector: ``\nabla\times\nabla{f}=\vec{0}``

* The divergence of a curl is ``0``: ``\nabla\cdot(\nabla\times F)=0``


[Helmholtz]() decomposition theorem says a vector field (``n=3``) which vanishes rapidly enough can be expressed in terms of ``F = -\nabla\phi + \nabla\times{A}``. The left term will be irrotational (no curl) and the right term will be incompressible (no divergence).




## Integrals


The definite integral, ``\int_a^b f(x) dx``, for a bounded univariate function is defined in terms Riemann sums, ``\lim \sum f(c_i)\Delta{x_i}`` as the maximum *partition* size goes to ``0``. Similarly the integral of a bounded scalar function ``f:R^n \rightarrow R`` over a box-like region ``[a_1,b_1]\times[a_2,b_2]\times\cdots\times[a_n,b_n]`` can be defined in terms of a limit of Riemann sums. A Riemann integrable function is one for which the upper and lower Riemann sums agree in the limit. A characterization of a Riemann integrable function is that the set of discontinuities has measure ``0``.


If ``f`` and the partial functions (``x \rightarrow f(x,y)`` and ``y \rightarrow f(x,y)``) are Riemann integrable, then Fubini's theorem allows the definite integral to be performed iteratively:

```math
\iint_{R\times S}fdV = \int_R \left(\int_S f(x,y) dy\right) dx
= \int_S \left(\int_R f(x,y) dx\right) dy.
```

The integral satisfies linearity and monotonicity properties that follow from the definitions:


* For integrable ``f`` and ``g`` and constants ``a`` and ``b``:
```math
\iint_R (af(x) + bg(x))dV = a\iint_R f(x)dV + b\iint_R g(x) dV.
```

* If ``R`` and ``R'`` are *disjoint* rectangular regions (possibly sharing a boundary), then the integral over the union is defined by linearity:

```math
\iint_{R \cup R'} f(x) dV = \iint_R f(x)dV + \iint_{R'} f(x) dV.
```


* As ``f`` is bounded, let ``m \leq f(x) \leq M`` for all ``x`` in ``R``. Then

```math
m V(R) \leq \iint_R f(x) dV \leq MV(R).
```

* If ``f`` and ``g`` are integrable *and* ``f(x) \leq g(x)``, then the integrals have the same property, namely ``\iint_R f dV \leq \iint_R gdV``.

* If ``S \subset R``, both closed rectangles, then if ``f`` is integrable over ``R`` it will be also over ``S`` and, when ``f\geq 0``, ``\iint_S f dV \leq \iint_R fdV``.

* If ``f`` is bounded and integrable, then ``|\iint_R fdV| \leq \iint_R |f| dV``.




In two dimensions, we have the following interpretations:

```math
\begin{align}
\iint_R dA &= \text{area of } R\\
\iint_R \rho dA &= \text{mass with constant density }\rho\\
\iint_R \rho(x,y) dA &= \text{mass of region with density }\rho\\
\frac{1}{\text{area}}\iint_R x \rho(x,y)dA &= \text{centroid of region in } x \text{ direction}\\
\frac{1}{\text{area}}\iint_R y \rho(x,y)dA &= \text{centroid of region in } y \text{ direction}
\end{align}
```

In three dimensions, we have the following interpretations:


```math
\begin{align}
\iint_VdV &= \text{volume of } V\\
\iint_V \rho dV &= \text{mass with constant density }\rho\\
\iint_V \rho(x,y) dV &= \text{mass of volume with density }\rho\\
\frac{1}{\text{volume}}\iint_V x \rho(x,y)dV &= \text{centroid of volume in } x \text{ direction}\\
\frac{1}{\text{volume}}\iint_V y \rho(x,y)dV &= \text{centroid of volume in } y \text{ direction}\\
\frac{1}{\text{volume}}\iint_V z \rho(x,y)dV &= \text{centroid of volume in } z \text{ direction}
\end{align}
```


To compute integrals over non-box-like regions, Fubini's theorem may be utilized. Alternatively, a **transformation** of variables



### Line integrals

For a parameterized curve, ``\vec{r}(t)``, the **line integral** of a scalar function between ``a \leq  t \leq b`` is defined by: ``\int_a^b f(\vec{r}(t)) \| \vec{r}'(t)\| dt``. For a path parameterized by arc-length, the integral is expressed by ``\int_C f(\vec{r}(s)) ds`` or simply ``\int_C f ds``, as the norm is ``1`` and ``C`` expresses the path.

A Jordan curve in two dimensions is a non-intersecting continuous loop in the plane. The Jordan curve theorem states that such a curve divides the plane into a bounded and unbounded region. The curve is *positively* parameterized if the the bounded region is kept on the left. A line integral over a Jordan curve is denoted ``\oint_C f ds``.

Some interpretations: ``\int_a^b \| \vec{r}'(t)\| dt`` computes the *arc-length*. If the path represents a wire with density ``\rho(\vec{x})`` then ``\int_a^b \rho(\vec{r}(t)) \|\vec{r}'(t)\| dt`` computes the mass of the wire.

The line integral is also defined for a vector field ``F:R^n \rightarrow R^n`` through ``\int_a^b F(\vec{r}(t)) \cdot \vec{r}'(t) dt``. When parameterized by arc length, this becomes ``\int_C F(\vec{r}(s)) \cdot \hat{T} ds`` or more simply ``\int_C F\cdot\hat{T}ds``. In dimension ``n=2`` if ``\hat{N}`` is the normal, then this line integral (the flow) is also of interest ``\int_a^b F(\vec{r}(t)) \cdot \hat{N} dt`` (this is also expressed by ``\int_C F\cdot\hat{N} ds``).


When ``F`` is a *force field*, then the interpretation of ``\int_a^b F(\vec{r}(t)) \cdot \vec{r}'(t) dt`` is the amount of *work* to move an object from ``\vec{r}(a)`` to ``\vec{r}(b)``. (Work measures force applied times distance moved.)



A **conservative force** is a force field within an open region ``R`` with the property that the total work done in moving a particle between two points is independent of the path taken. (Similarly, integrals over Jordan curves are zero.)

The gradient theorem or **fundamental theorem of line integrals** states if ``\phi`` is a scalar function then the vector field ``\nabla{\phi}`` (if continuous in ``R``) is a conservative field. That is if ``q`` and ``p`` are points, ``C`` any curve in ``R``, and ``\vec{r}`` a parameterization of ``C`` over ``[a,b]`` that ``\phi(p) - \phi(q) = \int_a^b \nabla{f}(\vec{r}(t)) \cdot \vec{r}'(t) dt``.

If ``\phi`` is a scalar function producing a field ``\nabla{\phi}`` then in dimensions ``2`` and ``3`` the curl of ``\nabla{\phi}`` is zero when the functions involved are continuous. Conversely, if the curl of a force field, ``F``, is zero *and* the derivatives are continuous in a *simply connected* domain, then there exists a scalar potential function, ``\phi,`` with ``F = -\nabla{\phi}``.


In dimension ``2``, if ``F`` describes a flow field, the integral ``\int_C F \cdot\hat{N}ds`` is interpreted as the flow across the curve ``C``; when ``C`` is a closed curve ``\oint_C F\cdot\hat{N}ds`` is interpreted as the flow out of the region, when ``C`` is positively parameterized.

**Green's theorem** states if ``C`` is a positively oriented Jordan curve in the plane bounding a region ``D`` and ``F`` is a vector field ``F:R^2 \rightarrow R^2`` then ``\oint_C F\cdot\hat{T}ds = \iint_D \text{curl}(F) dA``.

Green's theorem can be re-expressed in flow form: ``\oint_C F\cdot\hat{N}ds=\iint_D\text{divergence}(F)dA``.

For ``F=\langle -y,x\rangle``, Green's theorem says the area of ``D`` is given by ``(1/2)\oint_C F\cdot\vec{r}' dt``. Similarly, if ``F=\langle 0,x\rangle`` or ``F=\langle -y,0\rangle`` then the area is given by ``\oint_C F\cdot\vec{r}'dt``. The above follows as ``\text{curl}(F)`` is ``2`` or ``1``. Similar formulas can be given to compute the centroids, by identifying a vector field with  ``\text{curl}(F) = x`` or ``y``.


### Surface integrals

A surface in ``3`` dimensions can be described by a scalar function ``z=f(x,y)``, a parameterization ``F:R^2 \rightarrow R^3`` or as a level curve of a scalar function ``f(x,y,z)``. The second case, covers the first through the parameterization ``(x,y) \rightarrow (x,y,f(x,y)``. For a parameterization of a surface, ``\Phi(u,v) = \langle \Phi_x, \Phi_y, \Phi_z\rangle``, let ``\partial{\Phi}/\partial{u}`` be the ``3``-d vector ``\langle \partial{\Phi_x}/\partial{u}, \partial{\Phi_y}/\partial{u}, \partial{\Phi_z}/\partial{u}\rangle``, similarly define ``\partial{\Phi}/\partial{v}``. As vectors, these lie in the tangent plane to the surface and this plane has normal vector ``\vec{N}=\partial{\Phi}/\partial{u}\times\partial{\Phi}/\partial{v}``. For a closed surface, the parametrization is positive if ``\vec{N}`` is an outward pointing normal. Let the *surface element* be defined by ``\|\vec{N}\|``.

The surface integral of a scalar function ``f:R^3 \rightarrow R`` for a parameterization ``\Phi:R \rightarrow S`` is defined by
```math
\iint_R f(\Phi(u,v))
\|\frac{\partial{\Phi}}{\partial{u}} \times \frac{\partial{\Phi}}{\partial{v}}\|
du dv
```

If ``F`` is a vector field, the surface integral may be defined as a flow across the boundary through

```math
\iint_R F(\Phi(u,v)) \cdot \vec{N} du dv =
\iint_R (F \cdot \hat{N}) \|\frac{\partial{\Phi}}{\partial{u}} \times \frac{\partial{\Phi}}{\partial{v}}\| du dv = \iint_S (F\cdot\hat{N})dS
```




### Stokes' theorem, divergence theorem


**Stokes' theorem** states that in dimension ``3`` if ``S`` is a smooth surface with boundary ``C`` -- *oriented* so the right-hand rule gives the choice of normal for ``S`` -- and ``F`` is a vector field with continuous partial derivatives then:

```math
\iint_S (\nabla\times{F}) \cdot \hat{N} dS = \oint_C F ds.
```

Stokes' theorem has the same formulation as Green's theorem in dimension ``2``, where the surface integral is just the ``2``-dimensional integral.

Stokes' theorem is used to show a vector field ``F`` with zero curl is conservative if ``F`` is continuous in a simply connected region.

Stokes' theorem is used in Physics, for example, to relate the differential and integral forms of ``2`` of Maxwell's equations.

----

The **divergence theorem** states if ``V`` is a compact volume in ``R^3`` with piecewise smooth boundary ``S=\partial{V}`` and ``F`` is a vector field with continuous partial derivatives then:

```math
\iint_V (\nabla\cdot{F})dV = \oint_S (F\cdot\hat{N})dS.
```

The divergence theorem is available for other dimensions. In the ``n=2`` case, it is the alternate (flow) form of  Green's theorem.

The divergence theorem is used in Physics to express physical laws in either integral or differential form.