Sec2Sess43

Unformatted notes below. Notes on my shorthand method
Session 43: Clearer Notation
Examples to explain further dependent and independent variable relationships:
Example: f(x,y) = x+y
pd(f)/pd(x) = 1
Example 2:
x = u
y = u+v
then f = x+y = 2u+v
pd(f)/pd(u) = 2
Therefore x=u but pd(f)/pd(x) <> pd(f)/pd(u) where <> is "not equals"
when keeping x constant, keeping u constant is the same thing
however when keeping y constant, keeping v constant is not the same thing
when keeping y constant if x is changed than v will also have to change so the sum of y = u + v stays the same. If v is constant but u changes than y will change.
Notations do not make explicit what is being kept constant
the reason pd(f)/pd(x) <> pd(f)/pd(u) is that changing x changes u but keeps y constant. changing u changes x but keeps v constant. rewritten: v = y-x.
For the future to clarify things it will be said explicitly what is wanted to be kept constant. A subscript will be used to notate what variable is being kept constant for clearer notation.
Notation example (pd(f)/pd(x))_y where y is subscript indicating the variable to be kept constant.
Explicitly specifying the constant variable is especially useful when working with problems where multiple variables are related (dependent) on each other. When taking partial derivatives any variables other than the designated constrained ones are allowed to vary.
An issue that can occur is that there can be no single answer to a problem when variables are not
independent. Multiple answers can conflict with each other and the "expression" (formula) has
"no definite meaning" (no specific solution).
If the constraint variables in the partial derivative are explicitly stated (variables that don't change) than exact solutions can be found because multiple variables related to each other (codependent) are avoided. The independent variables are specifically stated to find an exact solution (a definite meaning solution).
Example: w = x^2 + y^2 + z^2 . Find (pd(w)/pd(x))_y. x and y are independent variables. z is the dependent variable. In writing: "the partial derivative of w with respect to x, with y (resp. z) held constant". In the notation unlisted variables are all dependent variables
Partial derivatives can be found using either total differentials or alternatively with implicit 
differentiation and the chain rule. This can often involve solving for certain variables by representing them with other variables. The variables in the form of other variables can than be substituted into equations to find solutions.
Total differentials find all partial derivatives which may be more than needed. In problems where multiple partial derivatives are wanted it can be more efficient to use total differentiation.
To help solve for different variables using substitution and other methods differentials can be treated like vectors with the d(x), d(y), and d(z) playing the role of i,j,k. The vectors can then have algebraic operations applied to them such as adding, subtracting, multiplying, etc.
Examples and further info:
https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-c-lagrange-multipliers-and-constrained-differentials/session-43-clearer-notation/MIT18_02SC_MNotes_n1_2.pdf
When applying total differentials all constants can be set to 0 values in the differential equation and therefore can be eliminated.
If a problem gives a constraint equation that equation can be differentiated to solve for an individual variable. That equation that solves for a variable can then be substituted into the original problem equation to simplify the number of variables for solving it.