2726.sagews

Topics Covered in Math 20C

In this worksheet, we cover some of the main topics in Math 20C (Calculus and Analytic Geometry in Science and Engineering).  Don't worry if you don't understand the details of some of the derivations here; this is just for exposition, and we will return to these methods and examples throughout the quarter.

In a nutshell, in this course we will extend the methods from Math 20B (limits, differentiation, integration, ...) to multiple dimensions.  After all, cars, comets, ...  follow trajectories in 2d and 3d.

 

Chapter 12. Vector Geometry

a. In two dimensions

Imagine an airplane flying with constant speed along a straight line, while the wind is blowing at it from an angle.  How do we characterize the trajectory of the plane under the influence of the wind? 

v_airplane = vector((5, 0))
v_wind =     vector((1, 1))

g = v_airplane.plot(color='blue')
h = v_wind.plot(color='red')
(g + h).show(aspect_ratio=1)

v_combined = v_airplane + v_wind
k = v_combined.plot(color='green')

(g + h + k).show(aspect_ratio=1)

b. In three dimensions

Real airplanes, comets, etc. are not confined to two dimensions but trace out curves in three-dimensional space.  How do we describe these trajectories?

How do we describe the curvature, length, etc. of these objects?

t = var('t')

parametric_plot3d((cos(2*pi*t), sin(2*pi*t), t), (t, -3, 3))

Incidentally, this is what the trajectories of charged particles (e.g. protons or electrons) in a magnetic field look like.

c. From curves to surfaces

A quadratic equation in three variables determines what is known as a quadric surface. We won't study these objects in detail, but they form a nice class of examples that easily visualized.

(There is no reason to restrict to quadratic equations other than elegance and simplicity -- in fact, elliptic curves (despite the name, these are particular instances of cubic surfaces) play an important role in number theory, cryptography, and other areas.

# One-sheeted hyperboloid
var('x, y, z')
hyperboloid = x^2/2 + y^2/2 == z^2 + 1; hyperboloid

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, x^{2} + \frac{1}{2} \, y^{2} = z^{2} + 1

implicit_plot3d(hyperboloid, (x, -3, 3), (y, -3,3), (z, -3,3))

Hyperboloid structures arise throughout nature and engineering (cooling towers, Gaudi's cathedral) -- read more at http://en.wikipedia.org/wiki/Hyperboloid_structure.  In fact, a simplified model (de Sitter space) represents our universe as a hyperboloid in 4d spacetime.

Let's change the equation of our surface somewhat and see what happens:

hyperboloid

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, x^{2} + \frac{1}{2} \, y^{2} = z^{2} + 1

surface = x^2/2 + y^2/2 == -z^2 + 1; surface

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, x^{2} + \frac{1}{2} \, y^{2} = -z^{2} + 1

# We get a completely different surface!
implicit_plot3d(surface, (x, -2, 2), (y, -2,2), (z, -2,2))

If we want to have any chance of understanding curves and surfaces in 3d, we'll need to be able to

• understand functions of two variables (continuity, partial derivatives and rate of change in a given direction)
• approximate such functions by a tangent plane
• ...

 

Chapter 14.  Differentiation in Several Variables

There are lots of functions of several variables, not all of them necessarily coming from quadratic surfaces or geometry.  Luckily, the usual techniques from calculus (limits, differentiation, ...) can be extended to these objects.

Chapter 15.  Multiple integration

Just as the integral of a function of one variable represents the "area under the curve", double and triple integrals measure area and volume, respectively.  It will be a theorem that these double and triple integrals can be calculated by doing 2 or 3 single integrals in succession.

var('x, y, z')
cylinder = x^2 + y^2 == 1; cylinder

\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} + y^{2} = 1

# Surface area of a cylinder
implicit_plot3d(cylinder, (x, -2, 2), (y, -2,2), (z,-1,1))

# Cut open the cylinder and flatten it into a sheet of dimensions 2pi x 2
plot3d(lambda x, y: 0, (x, 0, 2*pi), (y, 0, 2))
# Surface area of this sheet: 4*pi

Now, let's reverse engineer this formula a little bit:

1. 2 pi is the length of a circle of radius r = 1;
2. 2 is the total length (along the z-direction of the cylinder).

Hence:

or formally:

Not all surfaces can be cut open and flattened in the plane, so we'll need a more general formula for areas.  Double integrals will be essential here.

Similar formulas, but this time in terms of triple integrals, exist to calculate the volume of solid objects.

sphere = x^2 + y^2 + z^2 == 1; sphere
implicit_plot3d(sphere, (x, -2, 2), (y, -2,2), (z, -2,2))

var('r, theta, phi')
integrand = r^2*sin(theta)

Think of this as a function of r, theta, and phi, and let's calculate the following sequence of integrals:

r_int     = integral(integrand, (r, 0, 1))
theta_int = integral(r_int, (theta, 0, pi))
phi_int   = integral(theta_int, (phi, 0, 2*pi))

phi_int

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{4}{3} \, \pi

Hmm, this is precisely the volume of a ball with radius 1.  Is this a coincidence, and more importantly, is there a similar formula to calculate other volumes?