Lecture 08    

Partial Derivatives

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 - Curves and Motion
  • HW4 due next Tues
  • Early warning: Midterm 1 Thursday, 2/20.

1-minute review

Scalar Fields

A scalar field $f:\RR^n \to \RR$ is a function on a multidimensional vector space. Level sets are the sets of all input points with a particular output.

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

\[\lim\limits_{\vec x \to \vec p} f(\vec x) = L\] means $|f(\vec x) - L|$ can be made arbitrarily small by making $|\vec x - \vec p|$ sufficiently small.
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 substituting $x = r \cos \theta$, $y = r\sin \theta$ 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.

Differentiation

First Attempt

We could try to naïvely define a derivative of a multivariable function. \[ f'(\vec a) = \lim\limits_{\vec x \to \vec a} \frac{f(\vec x) - f(\vec a)}{\vec x - \vec a}\]

Don't do this!

We could try to naïvely define a derivative of a multivariable function. \[ f'(\vec a) = \lim\limits_{\vec x \to \vec a} \frac{f(\vec x) - f(\vec a)}{\vec x - \vec a}\]

Reset

Better to start with a question a derivative can answer.

Walking along a route $\langle x(t), y(t) \rangle$ through a field with elevation $f(x,y)$, what slope do you encounter at each $t$?

Definition

Let $f(x,y)$ be a scalar field. The partial derivative of $f$ with respect to $x$ is

\[f_x(a,b) = \lim\limits_{h \to 0} \frac{f(a + h, b) - f(a,b)}{h}.\]
Definition

Let $f(x,y)$ be a scalar field. The partial derivative of $f$ with respect to $y$ is

\[f_y(a,b) = \lim\limits_{h \to 0} \frac{f(a, b+h) - f(a,b)}{h}.\]
Notation

All of these are equivalent.

\[f_x = \frac{\partial f}{\partial x} = \partial_x f = f^{(1,0)}\]

and there are many more.

Examples

Compute:

$\displaystyle \frac{\partial}{\partial x} \left( x^2y - \sin(x-2y) \right)$

$\displaystyle \frac{\partial}{\partial y} \left( x^2y - \sin(x-2y) \right)$

$\displaystyle \frac{\partial}{\partial z} \left( \frac{z^2 \tan^{-1}(\sqrt{x^2+1})}{\cosh(xy)} \right)$

Higher Order Derivatives

2nd Order Derivatives

Since the partial derivative of a function is a function, we can iterate the process. \[f_{xx} = \frac{\partial^2 f}{\partial x^2}\] \[f_{xy} = \frac{\partial^2 f}{\partial y \partial x}\] etc.

PDEs

Tbe heat equation gives a relation on the temperature $u$ as a function of position $x$ and time $t$. \[u_t = u_{xx}\]

Mixed Partials

A quantity like $\frac{\partial^2 f}{\partial x \partial y}$ is a little harder to wrap ones head around.

Compute all mixed partials of the following functions:

$f(x,y) = xy^3 - y \sin x$

$r(x,t) = \frac{x}{x+t}$

$u(p,q) = e^{-p\sqrt{q}}$

Clairaut's Theorem

If the relevant mixed partials exist and are continuous in the neighborhood of a particular position $\vec x_0$, then \[ \frac{\partial^2}{\partial x_i \partial x_j} f(\vec x_0) = \frac{\partial^2}{\partial x_j \partial x_i} f(\vec x_0) \] for all $i,j$.

Example

Compute the partial derivative $g_{zzxw}$ for \[g(w,x,y,z) = w^2x^3yz^2 + \sin\left(\frac{xy}{z^2}\right).\]

\[ g_{zzxw} = g_{wxzz} = 2w(3x^2)(2y)\]

Estimating Partials

We can estimate values of partial derivatives even when lacking a formula for the underlying function.

A contour plot with sparsely plotted level sets

Does the following increase or decrease the accuracy of the estimates?

A contour plot with densely plotted level sets

Learning Outcomes

You should be able to...
  • Express partial derivatives in limit form and compute them in arbitrary dimensions.
  • Verify Clairaut's Theorem in higher dimensions.
  • Interpret PDE's and verify (but not find) solutions.
  • Approximate partial derivatives given a contour plot and/or tabular data.