class: center, middle, inverse, title-slide .title[ # Chain Rule ] .subtitle[ ## Session 04 ] .author[ ### Alejandro Ucan ] .date[ ### 2024-02-18 ] --- # Goals for the Session: * Introduce the concept of change of variables. <br/><br/> * Define the chain rule. <br/><br/> * Apply the chain rule. <br/><br/> --- # Motivation: > Consider the function `\(W(T,v)=13.12+0.62T -11.37v^{0.16}+0.39Tv^{0.16}\)` that measures the body's thermal sensation where `\(T\)` is the temperature and `\(v\)` is the wind speed. <br/><br/> But, both `\(T\)` and `\(v\)` depend on the position of the city where we are: `$$T(x,y)=17+3xy$$` `$$v(x,y)=-x^2+y^2$$` <br/><br/> Using a substitution we can express the Thermal sensation in terms of position doing: `$$W(T(x,y),v(x,y))$$` --- # Change of variables > **Definition:** A change of variables for a function `\(f:\mathbb{R}^n\to \mathbb{R}\)` is a collection of `\(n-\)`functions `\(g_1,\cdots,g_n:\mathbb{R}^m\to\mathbb{R}\)` such that `$$f(g_1(x_1,\cdots, x_m),g_2(x_1,\cdots, x_m),\cdots,g_n(x_1,\cdots, x_m)).$$`$ -- #### Example 1: Consider the function `\(f(x,y)=xy\)` and the change of variable `$$x(u,v,w)=u+v+w\quad y(u,v,w)=u-v+w.$$` Compute the function `\(f\)` with its change of variable. -- `$$f(x(u,v,w),y(u,v,w))=(u+v+w)(u-v+w).$$` --- #### Example 2: Consider the function `\(f(x,y)=\ln(x\cos(y^2))\)` and the change of variable `$$x(u,v)=\sin(uv)\quad y(u,v)=e^{uv}.$$` Compute the function `\(f\)` with its change of variable. -- `$$f(x(u,v),y(u,v))=\ln(\sin(uv)\cos(e^{2uv}))$$` --- #### Example 3: Consider the function `\(f(x,y,z)=\sqrt{x^2+y^2+z^2}\)` and the change of variable `$$x(u,v)=\sin(u) \quad y(u,v)=\cos(u)\quad z(u,v)=uv.$$` Compute the function `\(f\)` with its change of variable. -- $$ f(x(u,v),y(u,v),z(u,v))=\sqrt{u^2v^2}$$ --- # Chain Rule > **Theorem:** Assume that `\(u:\mathbb{R}^n\to \mathbb{R}\)` is a function of `\(n-\)`variables and `\(g_1,\cdots,g_n:\mathbb{R}^m\to\mathbb{R}\)` are functions of `\(m-\)`variables. Then, the chain rule states that `$$\frac{\partial u}{\partial x_j}=\frac{\partial u}{\partial u}\frac{\partial g_2}{\partial x_j}+ \frac{\partial u}{\partial g_2}\frac{\partial g_2}{\partial x_j}+\cdots +\frac{\partial u}{\partial u}\frac{\partial g_n}{\partial x_j}$$` for `\(j=1,\cdots,m.\)` --- # How it looks like that chain rule?  --- #### Example 1: Compute the partial derivatives of the function `\(f(x,y)=xy\)` with the change of variable `$$x(u,v,w)=u+v+w\quad y(u,v,w)=u-v+w.$$` -- `$$f_u(u,v,w)=(u-v+w)+(u+v+w)$$` `$$f_v(u,v,w)=(u-v+w)-(u+v+w)$$` `$$f_w(u,v,w)=(u-v+w)+(u+v+w)$$` --- #### Example 2: Compute the partial derivatives of the function `\(f(x,y)=\ln(x\cos(y^2))\)` with the change of variable `$$x(u,v)=\sin(uv)\quad y(u,v)=e^{uv}.$$` -- $$ f_u(u,v)=\frac{1}{\sin(uv)}\cos(uv)v+\frac{-2e^{xy}\sin(e^{2uv})}{\sin(uv)\cos(e^{2uv})}ve^{uv}$$