What is different between field , Vector function and plane?

Question

What is different between field , Vector function and plane , they seem all shown with same equation: $ \ r \left( t \right) = \left\langle { f \left( t \right),g \left ( t \right) , h \left( t \right)} \right\rangle$

Answer 1

Almost everything you'll see in math is defined in terms of sets and functions. A function in mathematics has a very precise definition, but for now the following "informal" definition is good enough.

A function consists of three pieces of data, $(f,A,B)$, and we write this as $f:A \to B$. Here, $A$ and $B$ are sets, called the domain and target space/ codomain respectively, and $f$ is a "rule", which assigns to each $a \in A$, an element $f(a) \in B$.

One example is $f:\Bbb{R} \to \Bbb{R}$, $f(x) = x^2 \sin(x)$. Notice how I gave you three pieces of information; I told you the domain, the target space and the actual rule (for each real number $x$, $f(x)$ is the number obtained by squaring and then multiplying by the sine of the number).

So, the answer to your question "what is the difference between field and vector function" is that the difference is in what the domain and target space of the function are. For example a scalar field.:

Definition $1$.

Let $S\subseteq \Bbb{R}^n$ be a non-empty set. A scalar field on $S$ is by definition a function $f:S \to \Bbb{R}$.

By the way, this should not be confused with the concept of a field of scalars (or simply a field) as defined in abstract algebra. Next, for vector fields:

Definition $2$.

Let $S\subseteq \Bbb{R}^n$ be a non-empty set. A vector field on $S$ is by definition a function $f:S \to \Bbb{R}^n$.

Another name as opposed to a "vector field on $S$" is "a vector-valued function defined on $S$", or more specifically, "an $\Bbb{R}^n$-valued function defined on the set $S$". Anyway, there are several ways of saying the same thing, and it doesn't matter what you call it, as long as you know what it is you're talking about.

So, you see everything is a function. The only question you have to ask is "what is the domain and target space of the function". By the way, in more advanced math, we typically provide slightly more refined definitions, but I think for the moment, the definitions I presented here are sufficiently general, and they serve well enough to illustrate my main point that "everything" is a function, and that you just have to ask what is the domain and target space.

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Now, a plane, let's call it $\Pi$, in $\Bbb{R}^3$ is by definition a certain subset of $\Bbb{R}^3$. There are several ways of describing planes as you may have seen For example, \begin{align} \Pi = \{(x,y,z) = \mathbf{r} \in \Bbb{R}^3| \, \, \, \mathbf{r} \cdot \mathbf{n} = p\} \end{align} or equivalently, if we write $\mathbf{n} = (a,b,c)$, then \begin{align} \Pi &= \{(x,y,z) \in \Bbb{R}^3| \, \, ax + by + cz = p\} \end{align} Basically, a plane in $\Bbb{R}^3$ is a subset $\Pi$, and it turns out that in order to specify a plane, you can usually use an equation to describe it. For example, when we say something like:

consider the plane $3x + 2y + z = 89$, what we mean is to consider the set of points $(x,y,z)$ which satisfy that equation, i.e \begin{align} \Pi := \{(x,y,z) \in \Bbb{R}^3| 3x + 2y + z = 89\} \end{align}

Another way of describing a plane is to describe it as the image of a certain function. For example, consider the function $f: \Bbb{R}^2 \to \Bbb{R}^3$ defined by the rule \begin{align} f(s,t) &= (0,1,0) + s(1,3,5) + t(2,8,7) \\ &= (s+ 2t, 1 + 3s + 8t, 5s + 7t) \end{align} Then the image/range set of the function $f$ describes a plane. i.e we consider the set \begin{align} \Pi := \text{image}(f) \subset \Bbb{R}^3. \end{align} This is also a plane.

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Summary: A scalar field and vector field are both functions, the difference is in what the domain and target space are. A plane is a subset, and to tell me specifically what the plane is, there are several ways of doing so, but the main point is that a plane is a certain subset... how you choose to describe it isn't that important.

Answer 2

Lets start for the biggest concept:

Field: A field is a set with two operations which satisfies a set of properties. Examples of fields are $\mathbb{Q,R,R\setminus Q,C, Z_p}$ with $p$ a prime numbers, etc.

Vector function: This is a name for applications which arrival set has more than one dimension, i.e $\mathbb{R^2},\mathbb{R^3}$, etc. For example, the functions that gives for every person the name and date of birth. $f(person) = (name, birth)$

Plane: This is a geometric concept. It's an unlimited flat region. With two vectors $v,w$ you can define a plane as follows: $\{\alpha u + \beta v : \alpha,\beta\in\mathbb{K}\}$, where $\mathbb{K}$ is the field of the vector space associated to $v,w$.

If you need more information, feel free to ask in comments