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Assignment 7

Problems

Partial Differential Equations

Partial derivatives are the language in which much of modern science is written. We saw in the last portion of the course that vector valued differential equations are the right language to describe the motion of single particles: but what about quantites that depend on more than one variable?

A first example is simply waves on a string: when a guitar string is pulled taught, if you try to pluck it away from rest it pulls back on you - the farther you pull it away, the harder it pulls back. The amount a string curves away from its straight line equilibrium is captured (roughly) by its concavity. And so one simple model of string motion would say the bigger the concavity the faster it wants to “snap back”. Said more precisely

The acceleration of the string is proportional to its concavity

Writing this in math - if the string’s displacement at position \(x\) and time \(t\) is given by the function \(W(x,t)\), we have the partial differential equation below: \[\partial_{xx} W = \partial_{tt}W\]

A solution to a partial differential equation is a function \(W(x,t)\) which makes the equation true (much like a solution to an algebra equation like \(x^2=4\) is a number that makes it true, like \(x=2\).)

  • Verify that the function \(W(x,t)=\sin(x-t)\) is a solution to the wave equation with a calculation. Now graph it in Desmos (using \(t\) as a parameter with a slider). Describe what the solution looks like in a sentence.
  • The function \(W(x,t)=\sin(x-2t)\) does not satisfy the wave equation. Can you modify the wave equation to by multiply one of the sides by a constant to get a new partial differential equation that this does satisfy?

Tangents and Normals to Surfaces

You are working with a real-estate developer to plan new environmentally-friendly infrastructure for a community in the rolling hills of Marin county. One particular hill you are developing is well approximated by the graph of the quadratic function \[z=5-\frac{x^2}{4}-\frac{y^2}{8}\] For \((x,y)\in [-2,2]\times[-2,2]\).

Your job is to find the optimal placement along the hill for the collection of solar panels that will provide electricity for the development. Solar panels work best when the sun light shines directly perpendicularly onto their surface, so you are looking for the location on the hill where the sun hits it most perpendicularly.

Consulting a local meteorologist you learn the direction of the average noontime sun at your build site is \(\langle 1,0,-4\rangle\). At which point \((x,y,z)\) should we build the solar panel farm? To figure this out, follow the outline below.

Finding the optimal location on a hill to place solar panels requires understanding the sun’s direction.
  • At any point \((x,y)\) on the plane, can you use partial derivatives to find a vector which is perpendicular to the hill at that point?
  • Think about the vector you found: is it pointed upwards (towards the sky) or downwards (into the hill)? There are always two directions a normal vector can point, and either one is fine but its important to know which one you are working with in applications.
  • Our real goal is to find when the hill is perpendicular to the direction of the sun. For which \((x,y)\) is the vector you found parallel (or, anti-parallel) to this?

Linear and Quadratic Approximations to Surfaces

Recall in Calculus I that a critical point of a function \(f\) is a point where the derivative is zero. Such critical points come up when trying to optimize a function (find a maximum or minimum value), but you may remember that more information is needed: we use something like the second derivative test to help us distinguish between maxima, minima, and other things like inflection points. A similar story plays out in multiple variables, and we will build it up in class shortly. In this problem we will just get a quick preview.

Imagine you are once again working for an engineering firm from last week’s assignment, where your team is tasked with maximizing the efficiency \(E\) of a rocket engine by varying the nozzle size \(n\) and the fuel temperature \(t\). The physics of the problem are too difficult to write down an equation for \(E\) by hand, so instead you have to resort to asking a computer to fun a simulation. Your simulation is able to compute the value of \(E\) and its derivatives at different inputs. Here are two example outputs from the program:

The Point \((n,t)=(1,2)\) \[E(1,2)=3,\, E_n(1,2)=1,\, E_t(1,2)=0\] \[ E_{nn}(1,2)=4,\, E_{tn}(1,2)=3,\, E_{nt}(1,2)=3,\, E_{tt}(1,2)=0\]

The Point \((n,t)=(4,1)\) \[E(4,1)=8,\, E_n(4,1)=0,\, E_t(4,1)=0\] \[ E_{nn}(4,1)=-3,\, E_{tn}(4,1)=0,\, E_{nt}(4,1)=0,\, E_{tt}(4,1)=-5\]

Question 1: Use the first derivatives to explain why \((1,2)\) is neither a maximum or minimum of efficiency.

Question 2: At \((4,1)\) the first derivatives are zero. This means a linear approximation alone is not useful to us, so we are at a critical point. To try and figure out if it is a maximum, minimum of zero we can go beyond linear and try a quadratic approximation, the multivariable analog of a second-order Taylor series.

Write down the quadratic approximation to \(E\) based at \((4,1)\), as a function of \(n\) and \(t\). Think back to our study of quadratic surfaces: perhaps by looking at slices of this (perhaps at \(n=4\) and \(t=1\)?), can you tell if its a max, min or saddle?