5.1 Introducing Euler's Method

State Space: Represents all possible states in a system.
Change Arrows: At every point in this space, there is a vector indicating the direction of change.
Unique Curve: A unique curve exists that is tangent to these change arrows but is unknown (the "red curve").

Purpose of Calculus: Determining the equation of the red curve from the vector field's equations (X' = f(X, Y) and Y' = G(X, Y)).
Limitation: While achieving this is possible in some linear systems, it remains unattainable in most cases (99%).

Approximation Method: Since the red curve's formula cannot be known, approximation is viable.
Error Measurement: Following a change arrow for one time unit yields an error in approximation relative to the red curve.
Reducing Error: By following the change arrow for shorter durations (delta T), the error decreases correspondingly.
Analogy: Approaching a circle by taking smaller steps illustrates how smaller increments lead to more accurate approximations.

Method Overview: Starts at an initial condition, follows change arrows for small time increments (delta T) to approximate the red curve.
Process:
- Follow the change arrow from the starting point for a short time to reach a new point.
- At each new point, determine the updated change arrow and repeat the process.
Result: Produces a blue broken line made of short straight segments that approximate the red curve.
Mathematical Assurance: The smaller delta T becomes, the closer the blue line approaches the true red curve.

Broader Usage: Euler’s method applies to any model, regardless of complexity.
Accuracy: The accuracy of the approximation can be improved by decreasing delta T for finer steps.
Key Distinction: In calculus, delta T approaches zero for theoretical theorems, whereas in Euler's method, delta T remains small but finite.