Table of Contents

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Main ideas

  • Reinforces both the Master Formula and differentials.
  • Sets the stage for path-independence.


  • Some familiarity with differentials.
  • Familiarity with the gradient.


A brief derivation of the master formula from the expression for the differential of a function of two variables.


  • whiteboards and pens
  • valley transparency (master available here)
  • blank transparencies and pens


  • Call someone from each group to the board to draw both their path and $d\rr$ on the topo map and show how they found $d\rr$. Discuss the different methods used by different groups. The idea here is that on a curve $dy$ is related to $dx$. Students are being asked to find this relationship, and plug it into the general expression for $d\rr$.
    “Use what you know! Any (algebraically correct) method will work.”
  • Emphasize that $\grad h$ is a property of the hill, while $d\rr$ is a property of the curve. The point of the master formula is that it naturally separates the information in $dh$ into these quite different geometric ideas.
  • Have the class discuss why the answer to the second integral is in fact easy to find without integration.


In the Classroom

  • This lab is on the long side; don't plan to do anything else in a 50-minute period. The wrapup alone easily requires 20 minutes to do properly; you may wish to do part of it in a subsequent class period.
  • Some students may not realize that $(1,1)$ is on the given circle!
  • Ask the students if their level curves are equally spaced.
    (They shouldn't be.)
  • Initially assign each group one of the curves; groups which finish quickly can try other curves. The first curve, the circle, is qualitatively different from the others, and more difficult; see Section 11.2. Furthermore, the instructions do not uniquely determine the curve in this case — although the final answer is unaffected. You may wish to assign this curve to a strong group, or not let any group try the circle until they have first done one of the other curves.
  • Some students substitute the given point into the height function before computing the gradient! Perhaps asking for a sketch of $\grad h$ at several points rather than just one would discourage this.
  • Ensure that students reduce to one variable before integrating.
  • Emphasize that one can plug in the relationship between $x$ and $y$ either before or after computing the differential of $h$. Which choice is easiest depends on the circumstances; both will work.
  • In the next-to-last question, groups may need to be reminded that they need to plug in information about their curve in order to find $dh$. They should use the expression for the differential of $h$ as a function of either one or two variables, rather than the master formula (which should not be used until the last question).
  • Some students will realize that the integrals must be the same because of the master formula before ever trying to compute the second integral. Such students should be praised — but still encouraged to compute the second integral without using the master formula.
  • On the circle, some students go from $x^2+y^2=a^2$ directly to always has $d\rr = dx\,\ii+dy\,\jj$ (or a similar expression in other coordinate systems). We literally stomp our feet when insisting that students start problems involving $d\rr$ by writing down one of these expressions! A discussion of this point works well as part of the wrapup.
  • See the discussion of using transparencies for the hill activity
  • Emphasize that $\DS\Lint$ is a definite integral, and that $\DS\Lint 0\,dx=0$ (not 1).

Subsidiary ideas

  • The gradient is perpendicular to level curves.
  • Emphasize that $df=\Partial{f}{x}\,dx+\Partial{f}{y}\,dy$ is a coordinate-dependent expression for $df$, whereas writing $df=\grad f\cdot d\rr$ is coordinate independent.


  1. Consider the valley in this group activity, whose height $h$ in meters is given by $h={ x^2\over10}+{ y^2\over10}$, with $x$ and $y$ (and 10!) in meters. Suppose you are hiking through this valley on a trail given by \begin{eqnarray*} x=3t \qquad y=2t^2 \end{eqnarray*} with $t$ in seconds (and where “3” and “2” have appropriate units).
    1. Starting from the master formula, determine how fast you are climbing (rate of change of $h$) per meter along the trail when $t=1$. You may find it helpful to recall that $ds=|d\rr|$.
    2. Starting from the master formula, determine how fast you are climbing per second when $t=1$.

Essay questions

  • During this activity, you drew a gradient vector on a topographic map. Can you draw this vector to scale? Explain.
  • What properties of your path are needed to compute the integrals in this activity? To determine the answer?


  • Discuss the relationship between the master formula, the gradient, topographic maps, and path-independence.
  • Discuss the fundamental theorem for gradients, namely that the line integral of a gradient is just an obvious antiderivative. Relate this to the geometry, especially the existence of a topo map.
  • Many students will integrate the two pieces of $dh=2x\,dx+2y\,dy$ separately, without worrying about the path. What path is implicitly being used?
  • We strongly discourage students from inserting artificial signs into expressions such as $d\rr = dx\,\ii + dy\,\jj$. This forces $dy<0$, and in some cases also $dx<0$, so that one must integrate from $1$ to $0$. By all means discuss the alternative convention with students, which requires $dx$ and $dy$ to always be positive, and then forces one to insert (and keep track of) appropriate signs by hand.
  • Following this lab is a good time to introduce or review the proof, using the master formula, that the gradient is perpendicular to level curves and that it points in the direction of maximal increase.
  • A great followup to this activity is a discussion of what questions you can answer using the master formula.
  • It is immediately obvious in polar coordinates that these integrals do not depend on $\phi$, and hence are independent of path.

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