In this lesson, we'll discuss how Curve utilizes a mathematical technique called Newton's Method to compute the amount of tokens to be exchanged and liquidity. We'll also provide an overview of how Newton's method works. ## Curve v1 Newton's Method Let's start with the simple case of Uniswap v2. For Uniswap v2, the amount of tokens to be received from a swap and the liquidity can be calculated directly from a simple equation: ```javascript x * y = L ``` where x represents the token being swapped for, and y represents the token to be received. L represents the liquidity. For Uniswap v3, there is no easy equation to calculate x, y, or D. ## Newton's Method Newton's Method is a mathematical technique used to find the solution for an equation f(x) = 0. The goal is to find the value of x where the function f(x) will evaluate to zero. Here is the algorithm for Newton's Method: ```javascript x_n = x_0 while True: x_{n+1} = x_n - f(x_n) / f(x_n) if abs(x_{n+1} - x_n) <= some small number: return x_{n+1} x_n = x_{n+1} ``` ## Using Newton's Method Let's take a look at how Newton's Method is used in Curve to calculate y and D. For the following examples, we'll be using the equation for n = 2: **Newton's method to find y** ```javascript F(y) = x * D(x+y) + x * y - x * D(1/2)^2 ``` We use Newton's Method to find the value of y where f(y) = 0. **Newton's method to find D** ```javascript F(D) = x * D(x+y) + x * y - x * D(1/2)^2 ``` We use Newton's Method to find the value of D where f(D) = 0. In the next video, we'll go over how to implement these examples using Python.