Sec2Sess34
Unformatted notes below. Notes on my shorthand method
Chain rule with more variables!
w=f(x,y), where x=x(u,v) and y=y(u,v)
=f(x(u,v),y(u,v))
Question: pd(w)/pd(u), pd(w)/pd(v) in terms of pd(w)pd(x),pd(w)pd(y),x_u,x_v,y_u,y_v
dw = f(x)*d(x)+f(y)*d(y) = f(x)*(x_u*d_u+x_v*d_v)+f(y)*(y_u*d(u)+y_v*d(v))
= (f_x*x_u+f_y*y_u)*d_u+(f_x*x_v+f_y*y_v)*d_v
pd(f)/pd(u) = (f_x*x_u+f_y*y_u)*d_u pd(f)/pd(v) = (f_x*x_v+f_y*y_v)*d_v
pd(f)/pd(u) = pd(f)/pd(x)*pd(x)/pd(u)+pd(f)/pd(y)*pd(y)/pd(u) pd(f)/pd(v) = pd(f)/pd(x)*pd(x)/pd(v)+pd(f)/pd(y)*pd(y)/pd(v)
due to partial derivatives formulas can not be simplified in the way of cancelling out pd(x) and 1/pd(x) and same for y in the formulas. regular derivatives can have that simplification.
multiple chain rules can be used sequentially for formulas with that amount of complication in its variables.
Example: polar coordinates x=r*cos(theta) y=r*sin(theta)
f=f(x,y) pd(f)/pd(r) = pd(f)/pd(x)*pd(x)/pd(r)+pd(f)/pd(y)*pd(y)/pd(r)=f(x)*cos(theta)+f(y)*sin(theta)
Gradient Vector a function where its components are the partial derivatives of a function g=<f_x,f_y,f_z> it is a way to group partial derivatives
Note: either dependency graph or total differentials methods can be used to find partial derivatives of problems that have multiple variables, e.g., x,y,z