Calculus
1 Double integral
\[V \approx \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {f\left( {x_i^*,y_j^*} \right)\,\Delta {\kern 1pt} A} }\]
- Double integral on f(x,y) that is bounded by x in interval [a,b] and y in interval [c,d]
- Trivial example is a constant function f(x,y) = c, meaning the Double Integral is just the volume of a prism of (b-a)(d-c)c
Fubini’s theorem: Obvious just like swapping double summation we can swap double integrals
\[\iint\limits_{R}{{f\left( {x,y} \right)\,dA}} = \int_{{\,a}}^{{\,b}}{{\int_{{\,c}}^{{\,d}}{{f\left( {x,y} \right)\,dy}}\,dx}} = \int_{{\,c}}^{{\,d}}{{\int_{{\,a}}^{{\,b}}{{f\left( {x,y} \right)\,dx}}\,dy}}\]
If f(x,y) = g(x)h(y) meaning we can decompose the function of x and y to 2 functions g(x) and h(y) where x and y are independent, we can also decompose the double integral into single integrals
\[\iint\limits_{R}{{f\left( {x,y} \right)\,dA}} = \iint\limits_{R}{{g\left( x \right)h\left( y \right)\,dA}} = \left( {\int_{{\,a}}^{{\,b}}{{g\left( x \right)\,dx}}} \right)\left( {\int_{{\,c}}^{{\,d}}{{h\left( y \right)\,dy}}} \right)\]
A conceptual and non-calculational approach.
1.0.0.1 Functions
\[ FoodEaten :: DayOfWeek \rightarrow AmountOfFoodEaten \] \[FoodEaten(Sunday) = Alot \] \[FoodEaten(Monday) = morsels \]
Functions are equations in which we can meta-label the input(independent variable) and output(dependent variable) of our variables.
This means a single equation can result in multiple different functions depending on our meta-labelling.
Functions are equations with an extra contraint: Every valid input must have 1 and only 1 output.
Not all equations are functions such as the equation for a circle,
but all functions are equations.
Explicit functions can take the written form of f(x) = …
Implicit functions can only be written as an equation.
Both behave the same; implicit and explicit is a meta-level syntax detail.
1.0.0.2 Differential and derivatives
Derivatives of a function tells us how the change of an input from some input value affects the change of an output.
In programming terms, derivatives are wrappers that is also dependent on the inputs of the function it wraps.
Derivative of a function will output a function with the same input type.
Example: Derivative of a FoodEaten function will output a RateOfFoodEaten function.
The RateOfFoodEaten function takes inputs of the same type as FoodEaten which are DayOfWeek. \[ t :: DayOfWeek \] \[ D(FoodEaten(t)) = RateOfFoodEaten(t) \]
Below is the derivative of the FoodEaten function. \[ D(FoodEaten(Sunday)) = RateOfFoodEaten(Sunday) = faster pace \]
\[ D(FoodEaten(Monday)) = RateOfFoodEaten(Monday) = slower pace \]
1.0.0.3 Typical (Non-differential) Equations
\[ Food + Population = t + ConstantK * Weather\] Equations are relations between two or more variables.
Main Question in solving Equations - What are the points that fit this specification?
Specification in this context means the Equation.
The solution of equations will output a set of points. (We will come back to this later when comparing with Differential Equations)
Another way to put it:
Given this equation that shows a relationship of variables between values and values,
What are the possible ways we can design points that satisfy these contraints?
(Typically these points end up looking like some line or a function)
1.0.0.4 Differential Equations
\[ D(Population(t)) = ConstantA * (1 - \frac{Population(t)}{FoodEaten(t)}) * D(FoodEaten(t))\]
Main Question in solving Differential Equations - What are the functions that fit these specifications?
However, specification in the context of Differential equations means equations that also includes derivatives or rates.
Another way to put it :
Given this differential equation that shows a relationship of variables between values with values, values with rates, and rates with rates,
What are the possible ways we can design a function that satisfies these contraints?
1.1 Executive Summary
1.1.0.1 Notice Similarity
- Solution of Differential Equation Outputs a set of functions.
- Solution of Equations Outputs a set of points.
1.1.0.2 Implications
If we have complete understanding of how each component of a system behaves in relation to every other component in said system,
we can predict, given initial conditions, what the state of the system will be in the future.
Simply put, Differential equations lets us see the future (assuming differential wrt time).
1.1.0.3 ODE vs PDE
- ODE - single variable(time)
- time is typically our single variable differential
- Want to predict system state wrt to time (predict future)
- PDE - multiple variables(time,space,etc)
- time and space
- Want to predict system state wrt to time and space
- Heat Equation is a PDE that lets us predict state(temperature) on a surface at some point in time.
ODE’s are really 1 variable PDE’s
1.1.0.4 Vector fields
One can imagine solutions of a differential eq to be multiple functions that fit a vector field.
It’s easier to imagine if the vector field is static.
But static vector fields only exist when the system is at a steady state.
