Sec2Sess42
Unformatted notes below. Notes on my shorthand method
Session 42: Constrained Differentials
Non-independent variables
How to handle functions when the variables are related
For example in Physics when pressue, volume, and tempurature are changing
General equations: f(x,y,z) where g(x,y,z)=c where c is a contraint.
If g(x,y,z)=c than each variable can be solved for as an expression of the two others.
Example:
z = z(x,y) that is z as a funtion of x and y.
A question can be asked, what is the rate of change of z with respect to x keeping y constant.
That equates to pd(z)/pd(x), partial derivative of z with respect to x.
Using partial derivatives a variable that can be expressed (found, determined) by the value of other variables can have its value found. The other variable values are applied to a function that results in the value of the variable in question.
If equations for finding the values of variables can not be solved exactly than approximation methods can be applied.
Approximation formulas can at times be simpler than exact solution formulas also
Example:
x^2+yz+z^3=8 at point (2,3,1)
Take the differential:
2x*dx+z*dy+(y+3z^2)*dz
the result 8 can be set to 0 because it is a constant.
2x+y+(y+3z^2)z = 0
for points (2,3,1):
2*2x+1*y+(3+3)*z = 4x+y+6z = 0
Each variable can be expressed in terms of the others e.g., 4x+y=-6z, 4x+6z=-y
Expressing in terms of the other variables shows how each variable depends on the others.
If we want to view z=z(x,y) than move z to the other side:
dz = -1/6(4*dx+dy) from -6z=4x+y
therefore pd(z)/pd(x) (partial derivative of z with respect to x) is -4/6 = -2/3
pd(z)/pd(y) = -1/6
In general:
g(x,y,z)=c, then
d(g)=g_x*d(x)+g_y*d(y)+g_z*d(z)=0 where d() is derivative
That can be used to solve for any variable that is wanted to be expressed in terms of the
others. This is useful for variables that depend on other variables for their value. This
could be used to find solutions to problems and avoid variables conflicting with each
other such as in collinearity or multicollinearity issues.
Solve for dz:
dz = -g_x/g_z*dx-g_y/g_z*dy
So, for pd(z)/pd(x) set y=constant=0=d(y)
pd(z)/pd(x) = -g_x/g_z
Implicit differentiation uses variables as a function of other variables
(e.g. d(y^2)/d(x)=2y*d(y)/d(x).
Total differentiation is taking the complete derivatives of the function and its constraint to find solutions.
General method for both types of differentiation:
An equation, such as a constraint equation, can be solved for a variable (as a derivative) then that variable can be applied to the problem's equation (in derivative form) to find the derivative with respect to a variable in question.