# Lecture Notes

Below are the lecture notes from my course on multivariable calculus, as well as some handouts I give to students for some of the experiments in the introductory physics lab courses. Note that I update my notes each term, but this website may not link to the latest editions.

### Notes on MTH3015: Calculus III

Visualizing 3D Curves with WolframAlpha

Week 3: Vector-valued Functions; Curves

Week 4: Functions of Several Variables; Limits and Continuity

Week 5: Partial Derivatives; The Chain Rule

Week 7: Directional Derivative and Gradient

Week 8: Taylor Series and Extrema

Week 9: Double Integrals; Polar Integrals

Week 10: Triple Integrals; Cylindrical and Spherical Co-ordinates

Week 11: The Transformation Theorem

### Notes on IPL/CPS Physics Labs

Experiment 12: The Simple Pendulum

Experiment 13: Simple Harmonic Motion

**Notes on Experiment 14: Standing Waves**

Sujeet Akula

Error Analysis

Standing Waves in a String

For a given configuration with $n$ nodes, you have $n-1$ measurements of half-wavelengths, say, $\{\ell_i\}_{i=1}^{i=n-1}$. From here, you calculate the average, $\ell$, and the standard deviation $\sigma_\ell$. This gives the error that you propagate to error bars, $\delta \ell = \sqrt{\sigma_\ell^2 + \Delta_\ell^2}$, where $\Delta_\ell$ is the systematic error. A good estimate of the system error is to measure the width of a node in your standing waves. You can either measure the widths of each and average for each standing wave, or simply take the largest width (which will be the width of a node in the lowest $n$ standing wave). Since you will make a plot with $v_{\mathrm{string}}^2$, we will propagate the error to this. Note that the frequency is fixed at $f=120 \mathrm{Hz}$. \begin{align} \delta\lambda &= 2\delta \ell \\ \delta v_{\mathrm{string}} &= f\delta\lambda \\ \delta (v_{\mathrm{string}}^2) &= 2v_{\mathrm{string}}\delta v_{\mathrm{string}} \end{align} The uncertainty in tension is much simpler. Since $F_t = mg$, and we will not make multiple measurements of the mass, take $\delta m = \Delta_m$, and let $\Delta_m = 1\,\mathrm{gram}$. Then, $\delta F_t = g\times(1\,\mathrm{gram})$. (Of course you will adjust $\Delta_m$ numerically to suit your units.) Thus, you plot the tension $F_t = mg$ on the $y$-axis, and on the $x$-axis, you have $v_{\mathrm{string}}^2$, with $x$ error bars, $\delta (v_{\mathrm{string}}^2)$, and $y$ error bars, $\delta F_t$. Your $y$ error bars will be fixed for each point, while your $x$ error bars will typically grow with $v_{\mathrm{string}}$. Be sure to calculate the slope and error in slope using http://freeboson.org/slope; the slope is the linear mass density of the string.

Standing Waves in the Air

The analysis here is similar. You will have written down the water-levels that correspond to the maxima, which you find on the way up and the way down. So, for $n$ such maxima, say you have $\{(x_i^u,x_i^d)\}_{i=1}^{i=n}$. Since there is a time-delay associated with hearing the resonance, we will immediately average these quantities, without considering statistical error to produce $\{x_i : x_i = (x_i^u + x_i^d)/2\}$. From here, we should be able to produce a set of $n-1$ simple differences that give the half-wavelengths: $\{\ell_i : \ell_i = x_{i+1}-x_i\}_{i=1}^{i=n-1}$. This means that for each tuning fork, you should have the given frequency $f$, measurements of the half-wavelengths $\{\ell_i\}_{i=1}^{i=n-1}$, where $n$ varies with the frequency. This time, we will be plotting $\lambda$ against $1/f$, so the analysis is simple. From your data ${\ell_i}$, calculate the average $\ell$, and the standard deviation $\sigma_\ell$. Then, $\delta \ell = \sqrt{\sigma_\ell^2 + \Delta_\ell^2}$, and we can set $\Delta_\ell = 0.5\,\mathrm{cm}$. From here, $\lambda = 2\ell$ and $\delta \lambda = 2 \delta \ell$. Assume no uncertainty in the frequency $f$, and therefore no uncertainty in $1/f$, so that you can plot $\lambda$ on the $y$-axis and $1/f$ on the $x$-axis. Your error bars in $y$ will be $\delta \lambda$. Again, calculate the slope and slope uncertainty. The slope here is the speed of sound.