Differentiable vector-valued functions from Euclidean space
Okay, imagine you have a toy car. When you push the car, it moves in a certain direction and you can see how far it moved. Now, if you want to figure out how fast the car is moving, you need to know how far it moved over a certain amount of time.
Now, let's say we have a toy car that can move in three different directions - forwards/backwards, left/right, and up/down. We can track the car's movement using something called a vector. A vector is like an arrow that shows us both the direction and how far the car moved.
So imagine we have a toy car that can move in any direction and we want to keep track of its movement using vectors. We can represent the car's movement as a function that takes its current position as an input and returns a vector that describes how far it will move in the next moment. This function is called a vector-valued function because it returns a vector value instead of a single number like most functions do.
Now, let's say we want to figure out how fast the car is moving. In order to do that, we need to figure out how its position is changing over time. We can do this by taking the derivative of the function that describes the car's movement. The derivative is like a mathematical way of measuring how much something is changing over time.
So if we have a vector-valued function that describes how a toy car moves in space, and we take its derivative, we get a new vector-valued function that describes how fast the car is moving in space. This is important for things like calculating velocity and acceleration, which are ways of measuring how quickly something is moving and how quickly it is changing its speed.
So when we talk about differentiable vector-valued functions from Euclidean space, we're basically talking about a fancy way of tracking the movement of something in space using vectors and derivatives.
Now, let's say we have a toy car that can move in three different directions - forwards/backwards, left/right, and up/down. We can track the car's movement using something called a vector. A vector is like an arrow that shows us both the direction and how far the car moved.
So imagine we have a toy car that can move in any direction and we want to keep track of its movement using vectors. We can represent the car's movement as a function that takes its current position as an input and returns a vector that describes how far it will move in the next moment. This function is called a vector-valued function because it returns a vector value instead of a single number like most functions do.
Now, let's say we want to figure out how fast the car is moving. In order to do that, we need to figure out how its position is changing over time. We can do this by taking the derivative of the function that describes the car's movement. The derivative is like a mathematical way of measuring how much something is changing over time.
So if we have a vector-valued function that describes how a toy car moves in space, and we take its derivative, we get a new vector-valued function that describes how fast the car is moving in space. This is important for things like calculating velocity and acceleration, which are ways of measuring how quickly something is moving and how quickly it is changing its speed.
So when we talk about differentiable vector-valued functions from Euclidean space, we're basically talking about a fancy way of tracking the movement of something in space using vectors and derivatives.