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3.10 Vector and Scalar Fields

3 min readjanuary 6, 2023

K

Krish Gupta

Daniella Garcia-Loos

Daniella Garcia-Loos

K

Krish Gupta

Daniella Garcia-Loos

Daniella Garcia-Loos

Vector and Scalar Fields

Vector and scalar fields are a Calculus 3 topic. But, they are easy enough and important enough that we learn to look at and read basic vector and scalar graphs.

Before we get into the actual fields, let's review scalar vs. vectors!

Scalar vs. Vector

Scalar—quantities that are described by magnitude (a numerical value) alone.

Example: She is five feet tall

  • Distance and Speed are scalar quantities

Vector—quantities that are described by a size (magnitude) and a direction (ex. East, Up, Right, etc)

Example: The gas station is five miles west from the car

  • , , and are vector quantities

Vectors can also be represented by arrows, and the length of the arrow should represent the magnitude of the described quantity.

Here are some key points about vectors and scalars:

  • A vector is a quantity that has both magnitude and direction. Examples of vectors include , force, and .
  • A scalar is a quantity that has only magnitude. Examples of scalars include mass, temperature, and energy.
  • Vectors can be added and subtracted using vector addition and subtraction, which involves both the magnitude and direction of the vectors. Scalars can be added and subtracted using simple arithmetic.
  • Vectors can be multiplied by scalars using scalar multiplication, which changes the magnitude of the vector but not its direction.
  • Vectors can be represented graphically using arrow diagrams, with the length of the arrow representing the magnitude of the vector and the direction of the arrow representing the direction of the vector. Scalars are usually represented as single numbers.

Vector Fields Snapshot

Properties: Magnitude and Direction

Representation: Arrows (curves)

Quantity Commonly Used to Graph:

Direction: Represented by the arrow

Magnitude: Represented by the density of the curves and arrows

Here are some key points about vector fields:

  • A is a mathematical representation of a vector-valued function in space.

  • A assigns a vector to each point in space, which means it can be used to represent things like the force field around a magnet or the flow of a fluid.

  • The vectors in a can be plotted as arrows on a graph, with the direction and length of the arrow representing the direction and magnitude of the vector at that point.

  • Vector fields can be used to visualize and understand how quantities change over a given area or region. For example, a could be used to show how the wind changes as you move across a region, or how the temperature changes as you move through a room.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-TiNeoStsFvCj.png?alt=media&token=858ef2ad-3f54-4d4a-b466-d6bd328aa87a

Image from wikimedia.org

  • Two Point Charges
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-fYnJPmwW5tj3.1?alt=media&token=5208fbff-ed1c-45c8-9d6c-882fde2ab9a0

Image from Ck12.org

Scalar Fields Snapshot

Properties: Magnitude

Representation: Curves

Quantity Commonly Used to Graph:

Direction: None

Magnitude: Written on the curves or a key given to the side

Here are some key points about scalar fields:

  • A is a function that assigns a scalar value (a single number) to every point in space.
  • Scalar fields can be used to represent things like temperature, pressure, and density.
  • Scalar fields are often visualized using contour maps, which show lines of equal value. For example, a temperature might be represented using a contour map with lines showing areas of equal temperature.
  • Scalar fields can be used to understand and analyze how a particular quantity changes over a given area or region. For example, a could be used to show how the temperature changes as you move through a room, or how the density of a fluid changes as you move through it.
  • Scalar fields are often used in physics, engineering, and other fields to model and analyze physical phenomena.
  • Scalar fields can be combined with vector fields to create more complex models of physical systems. For example, a temperature could be combined with a to model the flow of a fluid.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-zWDJmD383hJo.png?alt=media&token=3018351c-412c-4a09-8a9e-b0b8a5b11c10

Borrowed from Wikimedia

Key Terms to Review (7)

Acceleration

: Acceleration measures how quickly an object changes its velocity over time. It can refer to either speeding up or slowing down, or changing direction.

Displacement

: Displacement refers to the change in position of an object from its initial point to its final point, taking into account both distance and direction.

Electric field

: An electric field refers to an invisible area surrounding an electrically charged object or particle, where other charged objects experience either attraction or repulsion forces.

Electric Potential

: Electric potential refers to the amount of electric potential energy per unit charge at a specific point in an electric field. It represents how much work would be done to move a positive test charge from infinity to that point.

Scalar Field

: A scalar field is a mathematical function that assigns only a scalar value (magnitude) to each point in space, without specifying any direction. It describes quantities like temperature or pressure.

Vector Field

: A vector field is a mathematical function that assigns a vector to each point in space. It describes the direction and magnitude of a physical quantity at every point.

Velocity

: Velocity refers to the rate at which an object changes its position in a specific direction. It is a vector quantity that includes both speed and direction.

3.10 Vector and Scalar Fields

3 min readjanuary 6, 2023

K

Krish Gupta

Daniella Garcia-Loos

Daniella Garcia-Loos

K

Krish Gupta

Daniella Garcia-Loos

Daniella Garcia-Loos

Vector and Scalar Fields

Vector and scalar fields are a Calculus 3 topic. But, they are easy enough and important enough that we learn to look at and read basic vector and scalar graphs.

Before we get into the actual fields, let's review scalar vs. vectors!

Scalar vs. Vector

Scalar—quantities that are described by magnitude (a numerical value) alone.

Example: She is five feet tall

  • Distance and Speed are scalar quantities

Vector—quantities that are described by a size (magnitude) and a direction (ex. East, Up, Right, etc)

Example: The gas station is five miles west from the car

  • , , and are vector quantities

Vectors can also be represented by arrows, and the length of the arrow should represent the magnitude of the described quantity.

Here are some key points about vectors and scalars:

  • A vector is a quantity that has both magnitude and direction. Examples of vectors include , force, and .
  • A scalar is a quantity that has only magnitude. Examples of scalars include mass, temperature, and energy.
  • Vectors can be added and subtracted using vector addition and subtraction, which involves both the magnitude and direction of the vectors. Scalars can be added and subtracted using simple arithmetic.
  • Vectors can be multiplied by scalars using scalar multiplication, which changes the magnitude of the vector but not its direction.
  • Vectors can be represented graphically using arrow diagrams, with the length of the arrow representing the magnitude of the vector and the direction of the arrow representing the direction of the vector. Scalars are usually represented as single numbers.

Vector Fields Snapshot

Properties: Magnitude and Direction

Representation: Arrows (curves)

Quantity Commonly Used to Graph:

Direction: Represented by the arrow

Magnitude: Represented by the density of the curves and arrows

Here are some key points about vector fields:

  • A is a mathematical representation of a vector-valued function in space.

  • A assigns a vector to each point in space, which means it can be used to represent things like the force field around a magnet or the flow of a fluid.

  • The vectors in a can be plotted as arrows on a graph, with the direction and length of the arrow representing the direction and magnitude of the vector at that point.

  • Vector fields can be used to visualize and understand how quantities change over a given area or region. For example, a could be used to show how the wind changes as you move across a region, or how the temperature changes as you move through a room.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-TiNeoStsFvCj.png?alt=media&token=858ef2ad-3f54-4d4a-b466-d6bd328aa87a

Image from wikimedia.org

  • Two Point Charges
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-fYnJPmwW5tj3.1?alt=media&token=5208fbff-ed1c-45c8-9d6c-882fde2ab9a0

Image from Ck12.org

Scalar Fields Snapshot

Properties: Magnitude

Representation: Curves

Quantity Commonly Used to Graph:

Direction: None

Magnitude: Written on the curves or a key given to the side

Here are some key points about scalar fields:

  • A is a function that assigns a scalar value (a single number) to every point in space.
  • Scalar fields can be used to represent things like temperature, pressure, and density.
  • Scalar fields are often visualized using contour maps, which show lines of equal value. For example, a temperature might be represented using a contour map with lines showing areas of equal temperature.
  • Scalar fields can be used to understand and analyze how a particular quantity changes over a given area or region. For example, a could be used to show how the temperature changes as you move through a room, or how the density of a fluid changes as you move through it.
  • Scalar fields are often used in physics, engineering, and other fields to model and analyze physical phenomena.
  • Scalar fields can be combined with vector fields to create more complex models of physical systems. For example, a temperature could be combined with a to model the flow of a fluid.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-zWDJmD383hJo.png?alt=media&token=3018351c-412c-4a09-8a9e-b0b8a5b11c10

Borrowed from Wikimedia

Key Terms to Review (7)

Acceleration

: Acceleration measures how quickly an object changes its velocity over time. It can refer to either speeding up or slowing down, or changing direction.

Displacement

: Displacement refers to the change in position of an object from its initial point to its final point, taking into account both distance and direction.

Electric field

: An electric field refers to an invisible area surrounding an electrically charged object or particle, where other charged objects experience either attraction or repulsion forces.

Electric Potential

: Electric potential refers to the amount of electric potential energy per unit charge at a specific point in an electric field. It represents how much work would be done to move a positive test charge from infinity to that point.

Scalar Field

: A scalar field is a mathematical function that assigns only a scalar value (magnitude) to each point in space, without specifying any direction. It describes quantities like temperature or pressure.

Vector Field

: A vector field is a mathematical function that assigns a vector to each point in space. It describes the direction and magnitude of a physical quantity at every point.

Velocity

: Velocity refers to the rate at which an object changes its position in a specific direction. It is a vector quantity that includes both speed and direction.


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© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.