Chapter 16: Vector Calculus

16.1 Vector Fields

We can also write $\mathbf{F}$ as

Where $P,Q$ are scalar functions of two variables and are sometimes called scalar fields.

We can similarly define a vector field on $\mathbb{R}^{3},\mathbb{R}^{4},\dots$

16.2 Line Integrals

Consider a curve $C$ given by

Or, equivalently,

Assume $C$ is a smooth curve, i.e. $\mathbf{r}'$ is continuous and $\mathbf{r}'(t)=0$. Then, the line integral of a function $f(x,y)$ along $C$ can be written as the limit of a Riemann sum across the interval $[a,b]$.

Previously, we found that the arc length of a curve $C$ is

In other words (by deriving both sides with respect to $t$),

And thus,

Of course, the line integral does not depend on the choice of parametrization of the curve, as long as the curve is traversed exactly once.

Consider now a piecewise-smooth curve, i.e. $C$ is essentially several smooth curves continuously and smoothly connected together. Then the line integral on $C$ is simply the sum of the line integrals of each individual smooth sub-curve.

We may also take line integrals of a function $f(x,y)$ along $C$ with respect to either $x$ or $y$.

Frequently, line integrals with respect to $x$ and $y$ are summed. This is abbreviated as follows:

Note also that a given parametrization determines the orientation of a curve $C$ (basically, direction). Thus, $-C$ denotes the curve that is identical to $C$ but has opposite orientation. Then,

But, if one takes the line integral with respect to arc length, the value of the line integral does not change, i.e.

Because, of course, distance is positive.

We may similarly extend 2 dimensional line integrals into 3 dimensional space. In general, we can write the line integral with respect to arc length in $n$-dimensional space as follows:

And for each variable $x_{i}$ for $1\leq i\leq n$,

Now, consider a continuous vector field $\mathbf{F}$ defined on a smooth curve $C$. Let $\mathbf{F}$ be given by a vector function $r(t),\;a\leq t\leq b$. Then the line integral of $\mathbf{F}$ along $C$ is

Where $\mathbf{T}(t)=\frac{\mathbf{r}'(t)}{\lvert \mathbf{r}'(t) \rvert}$, which you may recall from [[#Curvature|the curvature section of 13.3]]. This integral is commonly used to calculate the work $W$ done by a force vector field $\mathbf{F}$, i.e.

In general, we can say

16.3 The Fundamental Theorem for Line Integrals

Recall the Fundamental Theorem of Calculus:

Where $F'$ is continuous on the interval $[a,b]$. We can similarly define the Fundamental Theorem for Line Integrals:

Now, consider two piecewise-smooth curves (commonly denoted paths) $C_{1},C_{2}$ with the same initial point $A$ and terminal point $B$. Then, by the fundamental theorem

Whenever $\nabla f$ is continuous. That is, the line integral of a conservative vector field depends only on the initial point and terminal point of a curve.

If $F$ is a continuous vector field with domain $D$, the line integral $\underset{ C }{\int} F\cdot \,\mathrm{d}\mathbf{r}$ is independent of path if the above is satisfied for any two paths $C_{1}$ and $C_{2}$.

The following theorem can be trivially shown.

Let an open domain $D$ be a domain such for every point $P\in D$, there is a disk with center $P$ that lies entirely in $D$. Additionally, let a connected domain $D$ be a domain such that any two points in $D$ can be connected via a path that lies entirely in $D$.

See the textbook for a proof of this theorem.

We do, however, still need to determine if a vector field $F$ is conservative. Suppose $F=P\mathbf{i}+Q\mathbf{j}$ is conservative, where $P,Q$ have continuous first-order partial derivatives. Then there exists a function $f$ such that $F=\nabla f$ such that

By Clairaut's Theorem

Thus,

The converse of this is only true for certain regions. Let a simple curve denote a curve that doesn't intersect itself anywhere between its endpoints. Then, let a simply-connected region $D$ be a region such that every simple closed curve in $D$ encloses only points that are in $D$. Intuitively, this means $D$ contains no holes and is one contiguous region. Then,

The ideas derived in this chapter can be applied to a continuous force field $F$ (i.e. physics).

Using the fact that $F(\mathbf{r}(t))=m\mathbf{r}''(t)$, where $\mathbf{r}(t)$ is the position function (this is the familiar equation $F=ma$), we can derive, from endpoints $a\to b$.

Or, in other words, the change in kinetic energy of the object.

Meanwhile, if we assume $F$ is a conservative force field, i.e. we can write $F=\nabla f$ for some function $f$, we can use the equation $P(x,y,z)=-f(x,y,z)$, where $P$ is the potential energy function. This naturally leads to $F=-\nabla P$, from which we easily derive, from endpoints $a\to b$.

Thus,

In other words, we just derived the Law of Conservation of Energy!

16.4 Green's Theorem

Green's Theorem helps relate a line integral around a simple closed curve $C$ and a double integral over the plane region $D$ bounded by $C$ (the interior of $C$, essentially).

Green's theorem is essentially the analogue of the Fundamental Theorem of Calculus for double integrals.

The proof is left out for brevity. As food for thought, consider that, to prove Green's Theorem when $D$ is a simple region, it's only necessary to show

Then, consider that proving Green's Theorem for a nonsimple region $D$ that is a finite union of simple regions is equivalent to dividing $D$ into simple regions and consider each simple region separately. (Hint: you will get some cancellation integrating along the boundary of the two simple curves).

For instance, consider trying to compute the area of a region $D$. We have that $A(D)=\underset{ D }{\iint} 1 \, \mathrm{d}A\implies \frac{ \partial Q }{ \partial x }-\frac{ \partial P }{ \partial y }=1$. There are many different possibilities for $P(x,y)$ and $Q(x,y)$ for this to be true. In general, we can write, by Green's Theorem,

Figuring out the values of $P(x,y)$ and $Q(x,y)$ that produce this formula is left as an exercise to the reader.

Recall the ideas given for the proof of Green's Theorem for nonsimple regions that are the finite union of simple regions. This idea frequently be applied to solve problems for such regions using Green's Theorem (i.e., dividing the nonsimple region into it simple regions and applying Green's Theorem individually).

16.5 Curl and Divergence

Curl

We begin with a discussion of the property of a vector field known as the curl. The curl vector is named as such because it describes the rotational motion of particles. Particles at $(x,y,z)$ in this vector field will tend to rotate around the axis aligned with the direction of the curl, and at a rate proportional to the curl vector's magnitude.

If $\mathbf{F}=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}$ is a vector field on $\mathbb{R}^{3}$ and the partial derivatives of $P,Q,R$ all exist, then the curl of $\mathbf{F}$ is the vector field on $\mathbb{R}^{3}$ defined by

It is easier to remember this using operator notation. We define the following as the vector differential operator $\nabla$:

We have seen this before in the notation of the gradient of some vector function $f$:

Knowing this, we may now write

There are a few nice facts about the curl of a vector field.

The proof of property $1$ is easy, and is left as an exercise to the reader. The proof of property $3$ require topics we have yet to discuss—in particular, Stokes' Theorem—and so will not be shown.

Note also the following ideas:

Divergence

The divergence of a vector field is again related to fluid dynamics. In particular, if $\mathbf{F}(x,y,z)$ is the velocity of a fluid, then its divergence is the net rate of change with respect to time of the mass of fluid flowing from point $(x,y,z)$ per unit volume. That is, it measures the fluid's tendency to diverge from the point.

If $\mathbf{F}=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}$ is a vector field on $\mathbb{R}^{3}$ and $\frac{ \partial P }{ \partial x },\frac{ \partial Q }{ \partial y },\frac{ \partial R }{ \partial z }$ all exist, then the divergence of $\mathbf{F}$ is described by

Note that the divergence of $\mathbf{F}$ is in fact a scalar field, and not a vector field like $\mathrm{curl\;}\mathbf{F}$.

We can also represent the divergence of $F$ in operator notation.

Since the curl of $\mathbf{F}$ is also a vector field on $\mathbb{R}^{3}$ if $\mathbf{F}$ is a vector field on $\mathbb{R}^{3}$, we can compute the divergence of $\mathrm{curl\;}\mathbf{F}$ as well. In fact,

The proof of this is fairly straightforward, and is left as an exercise to the reader.

Note also the following ideas:

We may also compute the divergence of a gradient vector field.

This expression occurs frequently, and is thus abbreviated as $\nabla^{2}f$. It is also denoted as the Laplace operator because of its relation to Laplace's Equation:

Vector Forms of Green's Theorem

Using the curl and divergence operators, we can rewrite Green's Theorem in vector form.

Curl

Consider the plane region $D$, its boundary curve $C$, and the functions $P,Q$ that satisfy the previously stated hypotheses of Green's Theorem. Consider also the vector field $\mathbf{F}=P\mathbf{i}+Q\mathbf{j}$. Then, its line integral is

Considering $\mathbf{F}$ as a vector field on $\mathbb{R}^{3}$ with third component $0$, we derive

That is,

Which allows us to express Green's Theorem in vector form:

That is, the line integral of the tangential component of $\mathbf{F}$ along $C$ is equivalent to the double integral of the vertical component of $\mathrm{curl\;}\mathbf{F}$ over the region $D$ enclosed by $C$.

Divergence

We can derive a similar formula involving the normal component of $\mathbf{F}$.

Let $C$ be given by

Then, we can write expressions for the unit tangent vector

and the outward unit normal vector

It follows that

Thus,

That is, the line integral of the normal component of $\mathbf{F}$ along $C$ is equal to the double integral of the divergence of $\mathbf{F}$ over the region $D$ enclosed by $C$.

16.6 Parametric Surfaces and Their Areas

Parametric surfaces are the analogues of parametric functions in 3 dimensions. They can be described by a vector function $\mathbf{r}(u,v)$:

This is a vector-valued function defined on a region $D$ in the $uv$-plane, and the parametric surface $S$ is created by $(u,v)$ varying throughout $D$. The equations

Are known as the parametric equations of $S$.

For a parametric surface $S$, there are two useful families of curves that lie on $S$: constant $u$ and constant $v$, i.e. the curves that correspond to the vertical and horizontal lines in the $uv$-plane. These curves are known as grid curves, which naturally follows from the curves' display in the $uv$-plane.

Surfaces of revolution can be expressed parametrically, e.g. the surface generated by rotating a curve $y=f(x)$ around the $x$-axis. Let $\theta$ be the angle of rotation of some point on the surface. Then, we can describe a point $(x,y,z)$ on $S$ as

It may help to verify this for yourself with an example function $f(x)$.

We can also describe the tangent planes to a parametric surface $S$. Let $S$ be traced out by the vector function

at a point $P_{0}$ with position vector $\mathbf{r}(u_{0},v_{0})$. Now, consider the grid curve $C_{1}$ lying on $S$ that occurs when $u$ is constant, i.e. $u=u_{0}$. Then, the tangent vector to $C_{1}$ at $P_{0}$ is obtained by taking the partial derivative of $\mathbf{r}$ with respect to $v$:

Similarly, when $v$ is kept constant, i.e. $v=v_{0}$, then the tangent vector to the grid curve $C_{2}$ at $P_{0}$ is

For a smooth surface, the tangent plane is the plane that contains these two tangent vectors, and $\mathbf{r}_{u}\times \mathbf{r}_{v}$ is a normal vector to the tangent plane.

Finally, we can consider calculating the surface area of a general parametric surface. The derivation is essentially a Riemann sum over the surface areas of sections of the tangent plane approximations of the surface, and thus will be skipped. The following theorem describes the formula succinctly:

Note that when we have $z=f(x,y)$, we can easily derive the surface area formula found in [[#15.5 Surface Area]] by considering the parametrization $x=x,\ y=y,\ z=f(x,y)$. That is,

16.7 Surface Integrals

Surface integrals are the the line integral equivalent in 3 dimensions. That is, surface integrals are to surface area as line integrals are to arc length.

We will soon define the surface integral of the function $f(x,y,z)$ over the surface $S$. Note that when $f(x,y,z)=1$, this is equivalent to the surface area of $S$. (Much like how the line integral of $f(x,y)=1$ is simply the arc length). We begin with parametric surfaces.

Parametric Surfaces

Suppose $S$ is described by

Again, we derive the formula for the surface integral via a Riemann sum. In this case, over infinitesimal subrectangles of the surface (specifically, taken from the tangent plane at each point in the surface), i.e.

Where $P_{ij}$ is the evaluation of $f$ at some point within the subrectangle and $\Delta S_{ij}$ is the area of the the subrectangle with indices $(i,j)$. Then, since $\Delta S_{ij}$ may be approximated by the area of a parallelogram in the tangent plane, i.e. $\Delta S_{ij}\approx \lvert \mathbf{r}_{u}\times \mathbf{r}_{v} \rvert\ \Delta u\ \Delta v$, where the tangent vectors at the point $P_{ij}$ are described by $\mathbf{r}_{u}=\frac{ \partial x }{ \partial u }\mathbf{i}+\frac{ \partial y }{ \partial u }\mathbf{j}+\frac{ \partial z }{ \partial u }\mathbf{k}$ and $\mathbf{r}_{v}=\frac{ \partial x }{ \partial v }\mathbf{i}+\frac{ \partial y }{ \partial v }\mathbf{j}+\frac{ \partial z }{ \partial v }\mathbf{k}$. Thus,

Note the similarity to the line integral formula: $\underset{ C }{\int} f(x,y,z) \,\mathrm{d}s=\int_{a}^{b} f(\mathbf{r}(t))\lvert \mathbf{r}'(t) \rvert \, \mathrm{d}t$.

Also note that if $S$ is a piecewise-smooth surface that intersect only on their boundaries, then it naturally follows that

Surface integrals may be applied to the same real-world ideas we have previously discussed. For instance, the total mass of a surface $S$ whose density is described by $\rho(x,y,z)$ is

And the center of mass is $(\overline{x},\overline{y},\overline{z})$, where

Moments of inertia, etc. can similarly be derived for surfaces.

Graphs of Functions

Now, consider a surface $S$ with equation $z=g(x,y)$. Then, it is a parametric surface with the following equations:

Thus,

And taking the cross product,

Substituting into our previously derived formula, we get

Oriented Surfaces

Like oriented line integrals, we have oriented surfaces. The purpose is to define surface integrals of vector fields.

Thus, we will consider only orientable (two-sided) surfaces.

Consider a surface $S$ that has a tangent plane at every point $(x,y,z)$ on $S$, disregarding boundary points. There exist two unit normal vectors $\mathbf{n}_{1},\mathbf{n}_{2}$ with $\mathbf{n}_{2}=-\mathbf{n}_{1}$ at $(x,y,z)$.

Choose one of the two unit normal vectors. Let $\mathbf{n}$ denote this chosen unit vector. Then, if $\mathbf{n}$ varies continuously over the entirety of $S$, then $S$ is an oriented surface and the given choice of $\mathbf{n}$ provides $S$ with an orientation. (Also, note that if your initial choice of $\mathbf{n}$ doesn't vary continuously over, the other choice of $\mathbf{n}$ wouldn't work either, since $\mathbf{n}_{2}=-\mathbf{n}_{1}$).

For a surface $S$ described by $z=g(x,y)$, we can describe $\mathbf{n}$ using the cross product of the tangent vectors described previously and repeated below for the reader's convenience.

This is a vector function that describes the normal vector at every point in the surface. So, it suffices to normalize this expression to find $\mathbf{n}$.

Note that the upward orientation of the surface is given by this expression since the $\mathbf{k}$-component is positive, and the downward orientation is given by $\mathbf{-n}$.

And if $S$ is a smooth orientable surface given parametrically by the vector function $\mathbf{r}(u,v)$, then it is simply

Surface Integrals of Vector Fields

Consider an oriented surface $S$ with unit normal vector $\mathbf{n}$. Now, let there exist a fluid with density $\rho(x,y,z)$ and velocity field $\mathbf{v}(x,y,z)$ that is flowing through $S$. The rate of flow per unit area may be described by $\rho \mathbf{v}$.

If we consider infinitesimal subsections of $S$ denoted $S_{ij}$, then $S_{ij}$ is approximately planar. Thus, the mass of fluid per unit time crossing $S_{ij}$ in the direction of the unit normal vector $\mathbf{n}$ is

Where $A(S_{ij})$ denotes the area of $S_{ij}$ and $\rho,\mathbf{v},\mathbf{n}$ are all evaluated at some point on $S_{ij}$. (For clarity, consider that $\rho \mathbf{v}\cdot \mathbf{n}$ is the magnitude of the component of the vector $\rho \mathbf{v}$ in the direction of the unit vector $\mathbf{n}$). By performing a Riemann sum over these infinitesimal $S_{ij}$'s, we get

And if we write $\mathbf{F}=\rho \mathbf{v}$, then $\mathbf{F}$ is a vector field on $\mathbb{R}^{3}$ and the integral becomes

This is a very common integral, particularly in physics, even when $\mathbf{F}$ is not $\rho \mathbf{v}$, and thus is denoted as the surface integral or flux integral of $\mathbf{F}$ over $S$. It's also called the flux of $\mathbf{F}$ across $S$.

When $S$ is parametrized by a vector function $\mathbf{r}(u,v)$, then

That is,

Meanwhile, if the surface $S$ is given by $z=g(x,y)$, then $x,y$ can be considered the parameters and we can write

Thus, we can write

Finally, let's consider some applications of the surface integral of a vector field to physics.

Consider an electric field $\mathbf{E}$. Then the surface integral $\underset{ S }{\iint} \mathbf{E}\cdot \, \mathrm{d}S$ is called the electric flux of $\mathbf{E}$ through the surface $S$. One important law of electrostatics is Gauss's Law, which states that the net charge enclosed by a closed surface $S$ is $Q=\epsilon_{0}\underset{ S }{\iint} \mathbf{E}\cdot \, \mathrm{d}S$, for some constant $\epsilon_{0}$ known as the permittivity of free space.

Consider now the scalar field $u(x,y,z)$ that describes the temperature at a point $(x,y,z)$ within a substance. Then the heat flow is defined by the vector field $\mathbf{F}=-K\ \nabla u$, where $K$ is the conductivity of the substance (and is typically experimentally determined). Then, the rate of heat flow across a surface $S$ in the substance is $\underset{ S }{\iint} \mathbf{F}\cdot \, \mathrm{d}S=-K\underset{ S }{\iint} \nabla u\cdot \, \mathrm{d}S$.