3 min read•january 6, 2023
Krish Gupta
Daniella Garcia-Loos
Krish Gupta
Daniella Garcia-Loos
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—quantities that are described by magnitude (a numerical value) alone.
Example: She is five feet tall
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
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:
Properties: Magnitude and Direction
Representation: Arrows (curves)
Quantity Commonly Used to Graph: Electric Field
Direction: Represented by the arrow
Magnitude: Represented by the density of the curves and arrows
Here are some key points about vector fields:
A vector field is a mathematical representation of a vector-valued function in space.
A vector field 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 vector field 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 vector field could be used to show how the wind velocity changes as you move across a region, or how the temperature changes as you move through a room.
Properties: Magnitude
Representation: Curves
Quantity Commonly Used to Graph: Electric Potential
Direction: None
Magnitude: Written on the curves or a key given to the side
Here are some key points about scalar fields:
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 min read•january 6, 2023
Krish Gupta
Daniella Garcia-Loos
Krish Gupta
Daniella Garcia-Loos
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—quantities that are described by magnitude (a numerical value) alone.
Example: She is five feet tall
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
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:
Properties: Magnitude and Direction
Representation: Arrows (curves)
Quantity Commonly Used to Graph: Electric Field
Direction: Represented by the arrow
Magnitude: Represented by the density of the curves and arrows
Here are some key points about vector fields:
A vector field is a mathematical representation of a vector-valued function in space.
A vector field 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 vector field 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 vector field could be used to show how the wind velocity changes as you move across a region, or how the temperature changes as you move through a room.
Properties: Magnitude
Representation: Curves
Quantity Commonly Used to Graph: Electric Potential
Direction: None
Magnitude: Written on the curves or a key given to the side
Here are some key points about scalar fields:
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.© 2024 Fiveable Inc. All rights reserved.
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