Why Do We Use Vector-Valued Functions in Multi-Dimensional Spaces?

[Muhammad Ali]

I am learning vector analysis these days. I have some knowledge of the applications of the vector field in the field of Engineering and Astrophysics. I am also aware of the fact that vector field is a function that assigns a unique vector to each point in two or three dimensional space.
Firstly, What I don't understand is the need for vector-valued function. For example, we know that a position vector determines the position of a point in two or three dimensional space and the vector field or vector-valued function assigns this vector. Therefore why we use vector-valued function when the solution (value of x & y) of ordinary function can be represented graphically which also represents a unique point.
Secondly, I can fully imagine the graphical representation of a function in three dimensions (to some extent if the fourth dimension is time) but I cannot imagine a n variable function graphically. What is the logic behind these sort of functions and is it possible to graph these functions and if yes how?
Finally, I am learning about the vector-valued function of more than three dimensions therefore how the vector field assigns a unique vector to the points generated by these vectors (keeping in mind that we have only three units vectors namely i, j and k along x, y and z axes)?
Please answer these questions very comprehensively because I am very confused by these questions.

 

[Office_Shredder]

The purpose of vector valued functions... let's say a particle is moving around a track. You want to know what its velocity is given what point on the track it's at. You would have a function that maps its position vector to its velocity vector, i.e. a vector field. Why WOULD you want to represent all of these functions graphically? It's not very rigorous, and it's not analytical, so you really can't do much besides smile at a pretty picture, which you can't even draw without getting the vector field first

Graphical representation, of course you can't imagine it. Humans inherently have difficulty imagining anything past three dimensions, because it's outside our scope. I don't see why you ask what the logic is behind the functions... the fact that we can describe them rigorously means that we can get around our lack of intuition, so when we come up with theories like relativity we're prepared to deal with more than three dimensions.

I don't get the last part, perhaps you can go into more detail?

 

[HallsofIvy]

If you are studying vector analysis why object to vector values functions?

Therefore why we use vector-valued function when the solution (value of x & y) of ordinary function can be represented graphically which also represents a unique point.

In a particular coordinate system, a vector can be identified with <x, y, z>. But the whole point of vectors is that they don't depend on the coordinate system. We use vectors rather than coordinates to avoid dependence on the coordinate system.

Secondly, I can fully imagine the graphical representation of a function in three dimensions (to some extent if the fourth dimension is time) but I cannot imagine a n variable function graphically. What is the logic behind these sort of functions and is it possible to graph these functions and if yes how?

Because there exist problems with "dimensionality" greater than 3! It is precisely because we cannot visualize (or graph) this situation that we want a "formalism" we can use instead.

Finally, I am learning about the vector-valued function of more than three dimensions therefore how the vector field assigns a unique vector to the points generated by these vectors (keeping in mind that we have only three units vectors namely i, j and k along x, y and z axes)?

No, I can't keep that in mind! If you are working in more than 3 dimensions, then you have more than 3 mutually orthogonal axes and so more than 3 basis vectors (which, I think, is what you meant rather than "unit vectors"- there are an infinite number of vectors of length 1).