Lecture 07    

Scalar Fields

APMA E2000

Drew Youngren dcy2@columbia.edu

$\gdef\RR{\mathbb{R}}$ $\gdef\vec{\mathbf}$ $\gdef\bv#1{\begin{bmatrix} #1 \end{bmatrix}}$ $\gdef\proj{\operatorname{proj}}$ $\gdef\comp{\operatorname{comp}}$

Announcements

  • Recitation 04 this week.
  • Quiz 3 this week - Curves and motion
  • HW4 due next Tues
  • Early warning: Midterm 1 Thursday, 2/20.

1-minute review

HW3.4 Solution
Curves

Let $\vec r(t), a \leq t \leq b$ be a smooth parametrization of a curve $\mathcal C$.

\[\begin{align*} \text{tangent vector}&:& \vec r'(t) \\ \text{unit tangent vector} &:& \vec T(t) = \frac{\vec r'(t)}{|\vec r'(t)|} \end{align*} \]
Curves

Let $\vec r(t), a \leq t \leq b$ be a smooth parametrization of a curve $\mathcal C$.

\[\begin{align*} \\ \text{arc length} &:& s = \int_a^t |\vec r'(u)|\,du \\ \text{curvature} &:& \kappa = \left| \frac{d\vec T}{ds} \right| = \frac{|\vec T'(t)|}{|\vec r'(t)|} \end{align*}\]

Scalar Fields

Definitions

A scalar field (or function of several variables) is simply a function of the type \[f:\RR^n \to \RR.\]

\[\operatorname{dom} f = \{(x,y)\in \RR^2: f(x,y)\in \RR\}.\]

\[\operatorname{im} f = \{f(x,y)\in \RR: (x,y)\in \operatorname{dom} f\}.\]

Examples
  • elevation: $f(x,y)$ is the height in feet above sea level for point at longitude $x$ and latitude $y$.
  • temperature: $u(x,y,z)$ is the temperature in the room at position $\langle x, y, z\rangle$.
  • $|\Psi(\vec x, t)|^2$ gives the probability density of finding a particle with wave function $\Psi$ at position $\vec x$ at time $t$.
A Note on Terminology

"Input vector, output scalar" is a little vague when it comes to implementation. One may have to coax one's data into the right form depending how the function is defined.

Level Sets

A level set for a scalar field $f$ is the set of all input values that give a particular output, $k$. \[ f^{-1}(k) = \{ \vec x \in \operatorname{dom} f : f(\vec x) = k\}\]

Example

The level sets of $f(x,y) = x^2 + y^2$ are circles.

Example

The level sets of $f(x,y,z) = x^2 + y^2 + z^2$ are spheres.

Examples

Find the domain, image, and sketch a few level curves for each function below.

  • $f(x,y) = (x^2 + 2 x y + y^2)/9$
  • $g(x,y) = e^{-x^2 - y^2}$
  • $h(x,y) = x \sin(y) / 2 + y\cos(2 x) / 2$
Exercises

Find the domain, image, and sketch a few level curves for each function below.

  1. $\displaystyle xy$
  2. $\displaystyle x \sin y$
  3. $\displaystyle \sqrt{4-x^2-y^2}$
  4. $\displaystyle \ln (x^2 + y^2)$
  5. $\displaystyle 2^{x-y}$

Graphs: 1 2 3 4 5

Limits

Good News

The definition of limit for a function of several variables needs barely be changed.

\[\lim\limits_{x \to a} f(x) = L\] means $|f( x) - L|$ can be made arbitrarily small by making $|x - a|$ sufficiently small.
\[\lim\limits_{\vec x \to \vec p} f(\vec x) = L\] means $|f(\vec x) - L|$ can be made arbitrarily small by making $|\vec x - \vec p|$ sufficiently small.

Continuity

Every elementary functions is continuous on its domain. That includes:

  • power functions $x^p$
  • absolute value $|x|$
  • exponents $a^{x}$
  • trigonometric functions $\sin(x), \cos(x)$
  • inverses of the above $\ln x$, $\arctan(x)$, etc.
  • and all sums, differences, products, quotients, and compositions thereof.
Example

\[ f(x,y,z) = \frac{\sin(x^2) - 3yz + \log_2 \frac{12 x z^2 - y}{\sqrt[3]{y^2 + z^2 + 7}}}{\tan^{-1}(|x|) + y^4z^2 + 4} \]

\[ \lim\limits_{(x,y,z) \to (0,-1,0)} f(x,y,z) \] \[ = -\frac14 \]

Bad News

Weird things can happen in several variables.

Key Example

Compute the following limit or show it does not exist. \[\lim\limits_{(x,y) \to (0,0)}\frac{xy}{x^2 + y^2} \]

Solution

The limit does not exist.

\[\lim\limits_{x \to 0} f(x,0) = \lim\limits_{x \to 0}\frac{x (0)}{x^2 + 0^2} = 0\]
\[\lim\limits_{y \to 0}f(0,y) = \lim\limits_{y \to 0}\frac{0y}{0^2 + y^2} = 0\] but...
\[\lim\limits_{x \to 0} f(x,x) = \lim\limits_{x \to 0}\frac{x(x)}{x^2 + x^2} = \frac12 \]
Showing Existence
What if all paths suggest the limit exists?
Key Example

Compute the following limit or show it does not exist. \[\lim\limits_{(x,y) \to (0,0)}\frac{x^2y}{x^2 + y^2} \]

Solution...

Polar coordinates

By substitutin $x = r \cos \theta$, $y = r\sin \theta$ we make the limit \[ \lim\limits_{(x,y) \to (0,0)} f(x,y) \] becomes

\[\lim\limits_{r \to 0^+} f(r \cos \theta, r \sin \theta). \]

Example

\[\lim\limits_{(x,y) \to (0,0)}\frac{x^2y}{x^2 + y^2} = \lim\limits_{r \to 0^+} \frac{(r^2 \cos^2 \theta)r \sin\theta}{r^2\cos^2\theta + r^2 \sin^2 \theta}\]

\[-1 \leq \cos^2 \theta \sin \theta \leq 1\]

\[\lim\limits_{r \to 0^+} -r \leq \lim\limits_{r \to 0^+} \frac{r^3 \cos^2 \theta \sin\theta}{r^2} \leq \lim\limits_{r \to 0^+} r\]

\[ \lim\limits_{r \to 0^+} r \cos^2 \theta \sin \theta = 0 \]

Rough Plan for Limits

  1. Check continuity. Try plugging in.
  2. Check a few paths. If any differ, limit DNE.
  3. Try polar coordinates. May need to translate to origin.
  4. Try something else. Squeezing, etc.

Learning Outcomes

You should be able to...
  • Identify domains and images of functions of several variables.
  • Match formulas, graphs, and contour plots of scalar fields.
  • Sketch level sets for basic functions of 2 variables.
  • Identify continuous functions and compute limits for functions of 2 variables.