<div dir="ltr">Hello Professor,<div><br></div><div>In Homework 4-(b), you let <img alt="v(t,y)=e^{\gamma_1 t}c(T-t, y+\gamma_2(T-t))" title="v(t,y)=e^{\gamma_1 t}c(T-t, y+\gamma_2(T-t))" class="va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09v(t,y)=e%5E{%5Cgamma%5F1%09t}c(T-t,%09y%2B%5Cgamma%5F2(T-t))" id="l0.1330293695632232" style="display: inline; vertical-align: -3.667px;" height="17" width="253">, i.e. treat <img alt="v" title="v" class="va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09v" id="l0.7288893704277106" height="7" style="display: inline;" width="7"> as a function of <img alt="y=log(x)" title="y=log(x)" class="va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09y=log(x)" id="l0.040101085995266406" style="display: inline; vertical-align: -3.667px;" height="16" width="73"> and <img alt="t" title="t" class="va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09t" id="l0.13244577908428656" height="10" style="display: inline;" width="5">. </div><div><br></div><div>I think what people usually do for this step is to regard <img alt="v" title="v" class="va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09v" id="l0.8304665379611256" style="display: inline;" height="7" width="7"> as a function of <img alt="t, z" title="t, z" class="va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09t,%09z" id="l0.32849826437793017" style="display: inline; vertical-align: -2.934px;" height="13" width="20">, with <img alt="z=y+\gamma_2(T-t)" title="z=y+\gamma_2(T-t)" class="va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09z=y%2B%5Cgamma%5F2(T-t)" id="l0.0017084878788211633" style="display: inline; vertical-align: -3.667px;" height="16" width="125">. And conclude that <img alt="v(t,z)" title="v(t,z)" class="va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09v(t,z)" id="l0.6277710443364419" height="16" style="display: inline; vertical-align: -3.667px;" width="41"> satisfies heat equation <img alt="\partial_t v = \kappa \partial_{zz}v" title="\partial_t v = \kappa \partial_{zz}v" class="va_li" src="https://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%5Cinline%09%5Cpartial%5Ft%09v%09=%09%5Ckappa%09%5Cpartial%5F{zz}v" id="l0.9913161387679212" height="14" style="display: inline; vertical-align: -2.2px;" width="85">.</div><div><br></div><div>With your notation, It appears to me that we actually didn't substitute any variable in this step, and I'm very confused with these notations. Could you explain?</div><div><br></div><div>Thanks!</div></div>