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We introduce the idea of a vector at every point in space.

### Types of functions

When we started on our journey exploring calculus, we investigated functions $f:\R \to \R$. Typically, we interpret these functions as being curves in the $(x,y)$-plane:

We’ve also studied vector-valued functions $\vec {f}:\R \to \R ^n$. We can interpret these functions as parametric curves in space: We’ve also studied functions of several variables $F:\R ^n \to \R$. We can interpret these functions as surfaces in $\R ^{n+1}$. For example if $n=2$, then $F:\R ^2\to \R$ plots a surface in $\R ^3$:

Now we are ready for a new type of function.

### Vector fields

Now we will study vector-valued functions of several variables: We interpret these functions as vector fields, meaning for each point in the $(x,y)$-plane we have a vector.

To some extent functions like this have been around us for a while, for if then $\grad G$ is a vector-field. Let’s be explicit and write a definition.
Consider the following table describing a vector field $\vec {F}$: What is $\vec {F}(1,7)$?
What is $\vec {F}(3,5)$?
What is $\vec {F}(2,6)$?
Consider the following picture: Which vector field is illustrated by this picture?
$\vec {F}(x,y)=\vector {x,y/2}$ $\vec {F}(x,y)=1/2$ $\vec {F}(x,y)=x+y/2$ $\vec {F}(x,y)=\vector {1,1/2}$

### Properties of vector fields

As we will see in the chapters to come, there are two important qualities of vector fields that we are usually on the look-out for. The first is rotation and the second is expansion. In the sections to come, we will make precise what we mean by rotation and expansion. In this section we simply seek to make you aware that these are the fundamental properties of vector fields.

Very loosely speaking a radial field is one where the vectors are all pointing toward a spot, or away from a spot. Let’s see some examples of radial vector fields.

Each of the vector fields above is a radial vector field. Let’s give an explicit definition.

Fun fact: Newton’s law of gravitation defines a radial vector field.

Is $\vec {F}(x,y,z) = \vector {x,y,z}$ a radial vector field?
yes no

Some fields look like they are expanding and are. Other fields look like the are expanding but they aren’t. In the sections to come, we’re going to use calculus to precisely define what we mean by a field “expanding.” This property will be called divergence.

#### Rotational fields

Vector fields can easily exhibit what looks like “rotation” to the human eye. Let’s show you a few examples.

At this point, we’re going to give some “spoilers.” It turns out that from a local perspective, meaning looking at points very very close to each other, only the first example exhibits “rotation.” While the second example looks like it is “rotating,” as we will see, it does not exhibit “local rotation.” Moreover, in future sections we will see that rotation (even local rotation) in three-dimensional space must always happen around some “axis” like this:

In the sections to come, we will use calculus to precisely explain what we mean by “local rotation.” This property will be called curl.

In this final section, we will talk about fields that arise as the gradient of some differentiable function. As we will see in future sections, these are some of the nicest vector fields to work with mathematically.

Let’s take a look at a gradient field.

Remind me, what direction do gradient vectors point?
Gradient vectors point to the maximum. Gradient vectors point up. Gradient vectors point in the initial direction of greatest increase.

### The shape of things to come

Now we present the beginning of a big idea. By the end of this course, we hope to give you a glimpse of “what’s out there.” For this we’re going to need some notation. Think of $A$ and $B$ as sets of numbers, like $A=\R$ or $A=\R ^n$ or $B=\R$ or $B=\R ^n$.

• $C(A,B)$ is the set of continuous functions from $A$ to $B$.
• $C^1(A,B)$ is the set of differentiable functions from $A$ to $B$ whose first-derivative is continuous.
• $C^2(A,B)$ is the set of differentiable functions from $A$ to $B$ whose first and second derivatives are continuous.
• $C^n(A,B)$ is the set of differentiable function from $A$ to $B$ where the first $n$th derivatives are continuous.
• $C^\infty (A,B)$ is the set of differentiable functions from $A$ to $B$ where all of the derivatives are continuous.

Here is a deep idea:

The gradient turns functions of several variables into vector fields.

We can write this with our new notation as:

Now we give a method to determine if a field is a gradient field.

Now try your hand at these questions:

Is $\vec {G} = \vector {2x+y^2,2y+x^2}$ a gradient field? If so find a potential function.
yes no
Is $\vec {G} = \vector {x^3,-y^4}$ a gradient field? If so find a potential function.
yes no
Find a potential function $F$ such that $F(\vec {0}) = 0$.
Is $\vec {G} = \vector {y\cos (x),\sin (x)}$ a gradient field? If so find a potential function.
yes no
Find a potential function $F$ such that $F(\vec {0}) = 0$.
Is $\vec {G} = \vector {\frac {-y}{x^2+y^2},\frac {x}{x^2+y^2}}$ a gradient field? If so find a potential function.
yes no