What is a Vector Field?

[Red_CCF]

Hi, I'm learning vector calc on my own and my book wasn't clear on some basic definitions.

1. My text mentions that a vector function r(t) = <f(t), g(t), h(t)>, where the f(t), g(t), h(t) are real valued functions. What exactly is a real-valued function? Like a function that produces a real number? If that's the case then we can't have a complex vector function?

2. What's the difference between a vector function and a vector field function? In my book all vector functions are of one variable and all vector field functions are of at least 2 variables; is this the difference?

3. What is a scalar field function? It looks exactly like a normal multi or single variable function.

4. How come vectors in a vector field are draw from the input (x,y) coordinate? I thought vectors can be moved so long its direction and magnitude are the same so why not just move all of them to start at the origin? Also, with wind vector fields etc. what rule prevents one from moving the vectors in these fields around which would screw up the analysis?

Thanks.

 

[mathman]

I'll try to answer some.
1. A real valued function is exactly what you said. Complex valued functions are used to define complex valued vectors. When you get to more advanced mathematics you will meet them.

3. Scalar is essentially a one component vector.

4. When adding vectors they need to be moved around.

 

[Red_CCF]

I'll try to answer some.
1. A real valued function is exactly what you said. Complex valued functions are used to define complex valued vectors. When you get to more advanced mathematics you will meet them.

3. Scalar is essentially a one component vector.

4. When adding vectors they need to be moved around.

What exactly is a one component vector?

 

[HallsofIvy]

I would NOT say that a scalar is a "one component vector"- vectors, even those with only one component, transform under change in coordinate system differently from scalars. A scalar is a number- a "scalar valued function" is a function that gives a single number as its value.

2) 2. What's the difference between a vector function and a vector field function?

A "vector function" is a function that, for each value of its argument, returns a vector. A "vector field function" returns a vector at each point of some geometric space. If your geometric space is R² (or R³) then, yes, it depends on two (or three) variables, the (x,y) (or (x,y,z)) coordinates of the point.

4. How come vectors in a vector field are draw from the input (x,y) coordinate? I thought vectors can be moved so long its direction and magnitude are the same so why not just move all of them to start at the origin? Also, with wind vector fields etc. what rule prevents one from moving the vectors in these fields around which would screw up the analysis?

They aren't necessarily. If you are given a vector of the form <a, b> (I prefer that notation to (a, b) which is easy to confuse with a vector) you can represent that vector by a directed line segment from $(x_0, y_0)$ to $(x_0+ a, y_0+ b)$ for any numbers $x_0$ and $y_0$. It just happens to be simplest to take $(x_0, y_0)= (0, 0)$

There is no rule that "prevents" you from moving vectors around, as long as both length and direction remain the same- and that will NOT "screw up" the analysis.

 

[Red_CCF]

Thanks for the response!

They aren't necessarily. If you are given a vector of the form <a, b> (I prefer that notation to (a, b) which is easy to confuse with a vector) you can represent that vector by a directed line segment from $(x_0, y_0)$ to $(x_0+ a, y_0+ b)$ for any numbers $x_0$ and $y_0$. It just happens to be simplest to take $(x_0, y_0)= (0, 0)$

There is no rule that "prevents" you from moving vectors around, as long as both length and direction remain the same- and that will NOT "screw up" the analysis.

But say for a gravitational vector field, I move all the vectors in the field to start at the origin (center of the Earth), wouldn't that change what the field was representing? Or what if I swap a few of the vectors such that the greatest magnitude was higher above the Earth and the smaller magnitude was closer to the Earth (which would indicate that gravity increases as you get farther away), based on the definition of the vector the two fields would still be the same (?) since I didn't change any single vector in the field just shifted them, but wouldn't that lead to the wrong conclusions when doing analysis?

A "vector function" is a function that, for each value of its argument, returns a vector. A "vector field function" returns a vector at each point of some geometric space. If your geometric space is R² (or R³) then, yes, it depends on two (or three) variables, the (x,y) (or (x,y,z)) coordinates of the point.

So a vector field function is a type of vector function that takes a point and returns a vector (which I can draw either starting from the input coordinate or at the origin (or at some other initial position))? Is there an example of a vector field function that takes a point in R² and produces a vector in R³ (so a point (x,y) gets turned into a vector <x,y,z>)?

 

[Mark44]

4. How come vectors in a vector field are draw from the input (x,y) coordinate? I thought vectors can be moved so long its direction and magnitude are the same so why not just move all of them to start at the origin? Also, with wind vector fields etc. what rule prevents one from moving the vectors in these fields around which would screw up the analysis?

The points (x, y) are the inputs to the vector-valued function, and the associated vectors are the outputs from this function. It wouldn't make any sense to move them around, since by doing so you would be changing the inputs but not the outputs.

 

[HallsofIvy]

Thanks for the response!



But say for a gravitational vector field, I move all the vectors in the field to start at the origin (center of the Earth), wouldn't that change what the field was representing? Or what if I swap a few of the vectors such that the greatest magnitude was higher above the Earth and the smaller magnitude was closer to the Earth (which would indicate that gravity increases as you get farther away), based on the definition of the vector the two fields would still be the same (?) since I didn't change any single vector in the field just shifted them, but wouldn't that lead to the wrong conclusions when doing analysis?

You are confusing "the vector field" with the "the vector". The gravitational force at a specific distance is a specific vector. You can move that vector around at will without changing its direction or magnitude. The fact that the vector field has vectors with greater magnitude closer to the Earth does not mean that a given vector increases magnitude as you move it.

So a vector field function is a type of vector function that takes a point and returns a vector (which I can draw either starting from the input coordinate or at the origin (or at some other initial position))?

Yes.

Is there an example of a vector field function that takes a point in R² and produces a vector in R³ (so a point (x,y) gets turned into a vector <x,y,z>)?

Sure. Take a vector field which, at each point on the surface of a sphere, returns a tangent vector at that point. If you want a concrete example, let f(x,y) be the vector giving wind velocity at each x= latitude, y= longitude on the Earth's surface.

 

[Red_CCF]

Sure. Take a vector field which, at each point on the surface of a sphere, returns a tangent vector at that point. If you want a concrete example, let f(x,y) be the vector giving wind velocity at each x= latitude, y= longitude on the Earth's surface.

But isn't a point on the surface of a sphere in (x,y,z) or R³?

You are confusing "the vector field" with the "the vector". The gravitational force at a specific distance is a specific vector. You can move that vector around at will without changing its direction or magnitude. The fact that the vector field has vectors with greater magnitude closer to the Earth does not mean that a given vector increases magnitude as you move it.

I meant that if I move a gravity vector that is initially farther from the planet, closer to it (without changing its magnitude and direction) and vice versa, would the new field be considered equal to the original? If so wouldn't that affect the any analysis done with the field?