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# Vector Fields

Gravitational fields, electric fields, magnetic fields — what exactly do we mean when we describe a physical phenomenon or process as a “field”? Lets discuss what a vector field is in math and physics, why these functions and graphs are such useful tools for modeling, and some of the most important applications of vector fields including gravitational fields, wind direction, and weather.

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## Vector Field Definition

In math and physics, a vector is a quantity with both magnitude and direction at a point somewhere in space. Forces, such as the force of gravity, are vectors — the amount of force exerted, as well as the direction it’s pointing, are important for solving physics problems. Velocity is also a vector; we need to know both the speed and direction of travel. A vector field is a function that describes what a vector looks like at many different points in space rather than just a single point.

A vector field is a mathematical function of space that describes the magnitude and direction of a vector quantity.

With a vector field equation for each dimension, we can plot a vector at any point$\left(x,y\right)$or$\left(x,y,z\right)$in real coordinate space. Vector fields can be visualized with graphs to show the magnitude and direction of vectors at many different position coordinates. We use vector fields to model and visualize many physical processes and phenomena, helping us understand the behavior of objects under a force like gravity.

Lets take a look at a common example of a vector field used in everyday life. A wind speed map constructed with real-world data is one application of a vector field used to understand the weather.

Vector field map of ocean wind velocity with Hurricane Ike visible in the Caribbean, L. Ricciardulli via Climate Data Guide, University Corporation for Atmospheric Research

In this vector field map, outlines of the continents are visible beneath a layer of arrows that represent the magnitude and direction of wind speed on the day of Hurricane Ike.1 A color map is also included to show wind velocity measurements, in units of. On this map, the dark blue areas show low to no wind activity, while bright pink and orange regions show much higher wind speeds. The arrows in the regions with higher wind velocities along the bottom edge of this map are noticeably longer, a visual indication of the greater magnitude of these vectors.

Recall the last time you watched the news or a weather channel — you probably have seen a vector map of the wind or other local weather patterns in a forecast of upcoming weather!

Graphs like these are useful tools to visualize and understand weather events like hurricanes, and even for visualizing and predicting upcoming weather. Next, lets take a closer look at some of the mathematical concepts behind vector fields and the most important applications in physics.

## Vector Field Equations

We defined a vector field as a function of space, which means there’s some math to be expected. Vector field equations can look many different ways — thats because each component function, or equation corresponding to a dimension in space, will have its own separate function. Remember that a vector pointing straight along (or parallel to) an axis is one-dimensional: vectors pointing left or right only have an$x$component, while vectors pointing up or down only have a$y$component.

Vectors in one dimension have one positional coordinate equal to zero and point straight horizontally or vertically, StudySmarter Originals

However, vectors that dont point straight will have an angle, which we measure from either the$x$or$y$axis. We can split a two-dimensional vector into its horizontal and vertical components, giving us two new vectors:

A two-dimensional vector can be broken into two separate one-dimensional component vectors, StudySmarter Originals

Vector field equations arent too different: we have$x$and$y$components, but instead of static points, we have functions where we can plug in different values. This lets us examine a vector quantity at any point of interest, as well as create vector field graphs. Lets put this together to write a general vector field equation:

,

where$f\left(x,y\right)$is the function for the$x$component and$g\left(x,y\right)$is the function for the$y$component. You might see this equation written with parenthesis instead of angle brackets, but both represent a vector field equation made up of component functions.

### Graphing Vector Fields

Lets go through an example using a 2D vector field equation.

We have the following vector field equation:

This breaks down into two component functions:$f\left(x,y\right)=x$for the$x$component, and$g\left(x,y\right)=-2y$for the$y$component. Now, how exactly do we use this vector equation? To start, we can pick evenly-spaced sample$\left(x,y\right)$points and plug these values into each vector component function. Although the vector field equation will tell us what a vector looks like at any set of real coordinates, we don’t want to plot too many points — that will make our graph too visually cluttered.

Let’s plug a couple of sample points into our above vector field equation.

 $\left(x,y\right)$ $\stackrel{\to }{r}\left(x,y\right)$ $\left(1,0\right)$ $\left(0,1\right)$ $\left(-1,0\right)$ $\left(0,-1\right)$ $\left(1,1\right)$ $\left(-1,1\right)$

And so on. The signs of our vector components tell us the direction, and remember that for vectors with two non-zero components, these vectors will have an angle associated with the direction, too. After finding enough vectors, you can plot them:

A portion of the vector field equation plotted with evenly spaced x and y sample values, StudySmarter Originals

This plot looks at a portion of our chosen vector field around the origin. We used a graphing calculator to help create this plot — although we can graph this by hand, graphing calculators are useful tools for accurately plotting vector fields.

Vector field functions can also be three-dimensional or even functions of time, but for now, well be sticking to two-dimensional functions and graphs.

To summarize, here are the most important points to remember about vector fields.

• Vector field equations are multivariable functions of 2D or 3D space, meaning an$x$,$y$, and sometimes$z$coordinate are needed.
• Vector field equations can also be a function of time.
• Vector field equations have a separate function for each component.
• To make a vector field graph, choose evenly-spaced sample coordinates and plug your chosen$\left(x,y\right)$values into each vector component function. Plot your vectors once you have enough to get an idea about the general shape of the vector field.
• Vector field graphs can also be created with a graphing calculator by providing the vector component functions. A graphing calculator will choose evenly-spaced sample points to plot.

## The Gravitational Vector Field

One of the most important physics applications of vector fields is modeling long-range forces. The gravitational vector field is the long-range force youre probably most familiar with. We call gravity a long-range force because it’s a weak force that dominates, or appears to be the primary acting force, at larger scales.

The gravitational vector field models the force that would be exerted due to gravity if we place a small mass at any point in the gravitational field of a larger mass.

Gravity is the universal attraction of all masses to one another, so we can model a gravitational field around any mass, whether its a moon, planet, or any smaller or larger object. Since the gravitational force is a vector field, we can choose many different points in space to examine how strong or weak the force of gravity would be on a smaller mass. To get a better understanding, let’s look at a couple of plots that visualize the gravitational vector field.

A portion of the 2D gravitational vector field and a 3D representation showing gravitational potential energy. The magnitude of the vectors near the central mass shows the greatest attractive force exerted, Creative Commons CC0 1.0

In this diagram, we have both a two-dimensional gravitational vector field plot and a three-dimensional gravity well. At the center of the$xy$plane, we have some mass with a gravitational field surrounding it — for example, this could be the Earth. The arrows represent the direction and magnitude of the gravitational force exerted on a smaller test mass — say a small rocky object, one that’s much smaller than the Earth. All vectors in the two-dimensional plot point directly towards the central mass: this is the attractive nature of gravity. Pay close attention to the magnitude of the vector arrows. The closer our small rocky object is to the Earth, the greater the gravitational force exerted. The strength of the force decreases as the distance between the two masses increases.

After examining the two-dimensional vector field plot, look at the visualization of the gravity well and check that this makes sense to you. Imagine that the Earth is centered at the bottom of this well — again, the gravitational force exerted on smaller masses is strongest in the space closest to the Earth. The shape of the gravity well emphasizes just how much stronger the pull of gravity is with decreased distance.

## The Gravitational Vector Field Equation

The gravitational vector field equation can be written using Newton’s law of universal gravitation:

$\stackrel{\to }{F}\left(r\right)=\frac{G{m}_{1}{m}_{2}}{{r}^{2}}$,

where$G$is the gravitational constant,${m}_{1}$and${m}_{2}$are the two masses, and$r$is the distance between the centers of the two masses. This equation gives us the gravitational force exerted by an object with mass${m}_{1}$on a second object with mass${m}_{2}$, such as the pull of the Earth on the moon. You might also see this written as:

$\stackrel{\to }{F}\left(r\right)\propto \frac{GM}{{r}^{2}}$,

where$M$is the mass of the object with the gravitational vector field we are modeling. This is a way of expressing that the strength of the gravitational field is inversely proportional to the squared distance from the central mass$M$.

## Types of Vector Fields

There are two types of vector fields in real two-dimensional space to consider. The first type is called a radial field.

A radial field is a vector field function where all vectors point directly towards or away from the origin. The magnitude of each vector is dependent on the vector’s distance from the origin.

Radial fields are rotationally symmetric, meaning the vector field will look the same after rotating the field about its center. Gravitational vector fields are an example of radial fields. All vectors point directly towards the origin, or center of mass, due to the attractive nature of gravity.

Gravitational attraction is an example of a radial vector field since all force vectors point toward the origin, Creative Commons CC0 1.0

The second type of vector field is called a rotational field or vortex field.

A rotational field is a vector field function where all vectors curl or swirl around the origin. The magnitude for each vector is dependent on the vector’s distance from the origin.

A rotational field can be used to model the flow of fluids or major weather events, like hurricanes.

Hurricanes, an example of a vortex field, show the swirling shape of rotational vector fields, Creative Commons CC0 1.0

## Vector Fields - Key takeaways

• A vector field is a multivariable function that models the magnitude and direction of a vector quantity at different points in 2D or 3D space.
• We use vector fields as a tool to understand, model, and predict physical processes, such as wind vectors and other weather patterns.
• A vector field equation can be broken into separate component functions, where each spatial dimension has its own function.
• We can plot vector fields by choosing evenly-spaced sample points to plug into each vector component function.
• Gravitational fields and other long-range forces can be modeled with a vector field function and visualized with vector field plots.
• Types of vector fields include radial fields, where all vectors point towards or away from the origin, and rotational fields, where vectors swirl or curl around the origin.
1. Ricciardulli, Lucrezia & National Center for Atmospheric Research Staff, Climate Data Guide, Feb 2017.

#### Flashcards in Vector Fields 13

###### Learn with 13 Vector Fields flashcards in the free StudySmarter app

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What are vector fields?

A vector field is a mathematical function that models the magnitude and direction of a vector quantity at different points in 2D or 3D space.

Gradient vector fields are found by applying the vector differential operator ∇ to a scalar function, which finds the partial derivative in each dimension. Gradient vector fields are also called conservative vector fields.

When is a vector field conservative?

A vector field is conservative when the line integral path is independent, meaning the curve we choose to integrate along will not change the final value as long as the endpoints remain the same.

Is the magnetic field a vector?

The magnetic field is a vector field that can be described with a vector field function and visualized with a vector field map. The direction and magnitude of the magnetic force at different points in space are modeled with the magnetic field.

Is an electric field a scalar or vector?

An electric field is a vector quantity, as the electric force has both magnitude and direction. The electric field models the electric force per unit charge in space.

## Test your knowledge with multiple choice flashcards

What is a vector field?

What is a component function?

True or false: vector fields can be functions of 2D space, 3D space, or time.

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