what is the difference between vector function and vector field ?
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A vector field means there's a vector defined everywhere in space. So like two examples of a vector field would be the wind velocity at every point in space. If you evaluate the point (x,y,z) then you'll get some vector telling you the wind wind speed and direction at that point. Another one would be the gradient of temperature, which tells you at every point which direction to go for the greatest increase in heat from where you're standing.
A vector function on the other hand might be something like how does your position vary over time? So you plug in a time and the vector function points to where you are.
A vector function takes in scalar as an input and outputs a vector.
t-> ( f(t), g(t), h(t) )
a vector outputs a vector at each point in the space
(x,y) -> ( f(x,y) , g(x,y) )
(x,y,z) -> ( f(x,y,z) , g(x,y,z) , h(x,y,z))
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There isn't really much difference I think, vector field is more like physically talking about a field of vectors, like a curve in space, we r talking about a bunch of vectors in space
A vector field is defined by a vector function, each vector function defines a particular field
just like using pairs of x and y you define a curve in space, using a vector function defines a vector field in space