# 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

**Lab Report Grading Structure**

Sujeet Akula

This is intended to serve as a more detailed breakdown of how your lab reports are graded, and precisely what information should be presented in your reports.

Introduction (10%)

In the introduction there are exactly crucial pieces that must be given:

- A brief summary of the basic theoretical concepts involved.
- The {\bf specific} derived quantities or relations that you will measure (e.g., the acceleration due to gravity).
- A description of what sort of raw data you will be taking and how this will relate to the final quantities or relations (e.g., we will carry out several measurements of the time for a pendulum to complete ten oscillations and use \( \dfrac{2\pi}{T} = \sqrt{\dfrac{g}{\ell}} \) to derive a measurement of $g$, the acceleration due to gravity).

Data (30%)

The portion of your grade attributed to "Data'' need not be a fully separate section of your report; the grade is based on the following pieces that will be present in several areas.

- All of your data and computed quantities must be organized into neat tables, with labels
- All dimensional quantities must have the the correct units attached. (If they are in a table, then the units of course may be given in the column/row header, but quantities given inline must have units attached to them.)
- All quantities deriving from your measurments must be presented with exactly 3 significant digits, except the uncertainty in the final quantitiy which should be given with only 2.
- All plots must occupy a full page in landscape mode
- All plot axes must be unambiguously labeled, with units. You should not label an axis on a plot as simply "\(\ell (m)\)'', instead you should have something like "Pendulum Length ($m$)'' unless the phrase would be very long, in which case a caption is necessary explaining the quantities.

Analysis (40%)

The parts of your lab report that contribute to your "Analysis'' grade are usually mixed with your "Data'' grade. This part of your grade is based on the following criteria.

- If error/uncertainty calculations are involved, this will constitute roughly {\bf half} of your "Analysis'' grade.
- All of the calculations based on raw data (completeness/correctness)
- Construction of the plots
- If the lab includes error/uncertainty, error bars are necessary on the plots.

Conclusion (10%)

Your conclusion should not include platitudes such as, "I learned a lot in this lab'' or "Overall, this lab was a success'' and definitely not "... human error ...''. Your conclusion {\bf should} contain the following:

- Explicit statement of the final measurements of quantities or relations mentioned in the introduction {\bf including uncertainty}, e.g. "We measured the acceleration due to gravity to be \( g = 9.75\,\pm\,0.15\,m/s^2 \), including all errors.'' (Note that $g$ is presented with 3 significant figures but $\delta g$ is given with 2.)
- A {\bf quantitative} comparison of your final measurement to what was expected.
- If your final measurement does not agree with what is expected, point out specifically what might have been wrong. E.g., "The large deviation from the accepted quantity \(9.81\,m/s^2\) is likely due our second data point which does not agree with the other data points. We may have accidentally timed 9 oscillations instead of 10 here.''

Post-lab Questions (10%)

The post-lab questions account for 10% of your lab report grade and are simply graded right/wrong.

Experiment 12: The Simple Pendulum