Vector Calculus I: Spring-2022
HW6 : Due Day 15 5/16

1. The Gas S0 4383S The temperature of a gas in \({}^\circ F\) is given by \(T=3x^2-5xy+2y^2z\), with \(x,y,z\) in feet.
   (What are the units of “\(2\)”, “\(3\)”, and “\(5\)”?)
   1. What is the rate of change in the temperature at the point \((1,2,3)\) in the direction of \(\boldsymbol{\vec v}=2\,\boldsymbol{\hat x}+\boldsymbol{\hat y}-2\,\boldsymbol{\hat z}\)?
      Give units!
   2. What is the direction of maximum rate of change of temperature at the point \((1,2,3)\)?
      What are the units?
   3. What is the maximum rate of change of temperature at the point \((1,2,3)\)?
      Give units!
2. The Path S0 4383S You are climbing a hill along the steepest path, whose slope at your current location is \(1\over5\). There is another path branching off at an angle of \(30^\circ\) (\(\pi\over6\)). How steep is it?
3. The Hill (wrapup) S0 4383S

   After completing the Hill activity, answer the following question:

   On the topographic map below, draw the path from point \(B=(4,-9)\) to the center (representing the top of the hill) that you believe represents the shortest path between those two point on the actual hill.

   Justify your choice by explaining your reasoning.