Name: work, circulation, and flux integrals
Uploaded: 2023-05-16T08:18:40.000Z
Duration: 1 min 15 s
Channel: JoVE

K12 Standards

Florida Department of Education

High School

Physical Science

SC.912.P.12

SC.912.P.12.1

Line, Surface, and Volume Integrals

A line integral is defined as the integral of the dot product of a vector function with an infinitesimal displacement vector carried out along a prescribed path.
For example, the line integral of a force vector gives work.
A surface integral is the integral of the dot product of a vector function with an infinitesimal area vector carried out over a prescribed surface.
The area vector is directed perpendicular to its surface.
Consider a vector function that is the product of the density and velocity of a fluid. The surface integral of this vector calculates its flux through the surface.
The line and surface integrals reduce to a closed-line or closed-surface integral when the path or surface is closed.
The integral of the product of a scalar function and an infinitesimal volume is the volume integral.
For example, the volume integral of the density function gives the total mass.
When the scalar function in the volume integral is replaced with a vector function, it reduces to a combination of integrals of scalar functions.

line, surface, and volume integrals

A line integral for a vector field is defined as the integral of the dot product of a vector function with an infinitesimal displacement vector along a prescribed path. If the prescribed path is closed, the integrals reduce to a closed-line integral. The closed-contour integral of the vector field is referred to in terms of the circulation of the vector field around the closed path. A vector with zero circulation around every closed path is called a conservative field, while one with non-zero circulation around a closed path is called a non-conservative field. For example, the line integral of a force along a prescribed path yields work. For a closed path, the net work done by the field or against the field is zero.
A surface integral is the integral of the dot product of a vector function with an infinitesimal area vector over a prescribed surface. The area vector is always directed perpendicular to its surface at its given location. The surface integral of a vector is the flux of this vector through the surface. If the prescribed path or surface is closed, the integrals reduce to a closed-surface integral. For a closed surface, the direction of the area vector points outward. For open surfaces, the direction depends on the contour enclosing the surface. The direction of the area vector is chosen so that it is always perpendicular to the surface and points outward from the region enclosed by the contour. According to the right-hand rule, if the fingers are along the direction of travel with the palm facing the interior of the surface, then the thumb points along the positive area vector. For example, consider a vector function that is the product of the density and velocity of a fluid. The surface integral of this vector can be used to calculate its flux through the surface.
Surface and line integrals of vector fields are more significant in physics than those of scalar fields because they have a direct physical interpretation related to the flow of energy or matter. Scalar fields are mathematical entities that represent physical quantities with only magnitude, such as temperature, pressure, or electric potential. A line integral of a scalar field is the integration of the scalar function along a path in the domain of the function, measuring the total contribution of the scalar function along the path. For example, the line integral of temperature along a path represents the total amount of heat gained or lost along the path. Similarly, a surface integral of a scalar field is the integration of the scalar function over a surface in the domain of the function, measuring the total contribution of the scalar function over the surface. For instance, the surface integral of fluid density over a surface represents the total amount of fluid passing through the surface. However, the direct physical interpretation of scalar field integrals is limited compared to that of vector field integrals, making the latter more important in physics.
The integral of the product of a scalar function and infinitesimal volume is the volume integral. For example, the volume integral of the density function gives the total mass. Similarly, the volume integral of the energy density represents the total energy stored in the volume. When the scalar function in the volume integral is replaced with a vector function, it reduces to a combination of integrals of scalar functions.

A line integral for a vector field is defined as the integral of the dot product of a vector function with an infinitesimal displacement vector along a ...

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Education

JoVE Core

Physics

Unit 1

Vectors and Scalars

KEY TERMS AND CONCEPTS

work, circulation, and flux integrals

Work, circulation, flux, and volume integrals describe how scalar and vector fields add up over a path, surface, or region. For high school science, these ideas help connect math to motion, flow, and stored mass or energy.

A line integral for a vector field adds the dot product of a vector function and a tiny displacement vector along a chosen path. If the path is closed, the result is a closed-line integral. The value around a closed path is called the circulation of the vector field.

A field with zero circulation around every closed path is called conservative. A field with non-zero circulation around a closed path is non-conservative. A line integral can also represent work. For a force acting along a path, the line integral gives the work done. For a closed path, the net work done by or against the field is zero.

A surface integral for a vector field adds the dot product of a vector function and a tiny area vector over a surface. The area vector is perpendicular to the surface at each point. The surface integral of a vector is the flux through the surface. If the surface is closed, the result is a closed-surface integral.

For a closed surface, the area vector points outward. For an open surface, its direction depends on the contour that encloses the surface. The direction is chosen so it stays perpendicular to the surface and points outward from the enclosed region. The right-hand rule helps set the positive area vector. If the fingers follow the direction of travel and the palm faces the interior, the thumb points along the positive area vector.

A fluid example helps show why surface integrals matter. If a vector function is the product of fluid density and velocity, its surface integral gives the flux through the surface. Surface and line integrals of vector fields are often more important in physics than those of scalar fields because they have direct meaning for the flow of energy or matter.

Scalar fields describe quantities with magnitude only, such as temperature, pressure, or electric potential. A line integral of a scalar field adds the scalar function along a path in its domain. For example, the line integral of temperature along a path represents the total heat gained or lost along that path. A surface integral of a scalar field adds the scalar function over a surface in its domain. For example, the surface integral of fluid density over a surface represents the total amount of fluid passing through it.

These scalar-field integrals have a more limited physical meaning than vector-field integrals. Volume integrals go one step further. They add a scalar function times a tiny volume element. The volume integral of density gives total mass. The volume integral of energy density gives total energy stored in the volume. When a vector function replaces the scalar function in a volume integral, the result becomes a combination of scalar integrals.

Work, Circulation, and Flux Integrals

introduction to scalars

Scalars in Everyday Physics

Scalar quantities are physical quantities that can be described completely by one number and a unit. A class period can last 50 min, a gas tank can hold 65 L, and the distance between two posts can be 100 m. In each case, the value is fully given with no direction needed. The word scalar is a synonym for number. Time, mass, distance, length, volume, temperature, and energy are all examples of scalar quantities. When two scalar quantities have the same units, they can be added or subtracted...

introduction to vectors

Vectors, Scalars, and Direction

Vectors and scalars describe physical quantities in different ways. Some quantities need both magnitude and direction to be fully specified. In those cases, the quantity is called a vector quantity. Examples include displacement, velocity, position, force, and torque. A quantity with only magnitude and no direction is a scalar quantity. The difference matters in everyday examples. If a rescue team knows only the distance to a distress signal, that is not enough to reach it quickly. They also...

vector components in the cartesian coordinate system

vector-components-in-the-cartesian-coordinate-system

Breaking Vectors into x and y Parts

Vectors in a Cartesian coordinate system are often split into x and y parts. This makes it easier to describe direction and size on a rectangular plane. It also matches the way people give directions in everyday life, such as east, west, north, south, and an angle of travel. In a plane, a point is written as an ordered pair, (x, y). A vector is described in a similar way with its vector coordinates. The x-coordinate of the vector is its x-component, and the y-coordinate is its y-component.

polar and cylindrical coordinates

Using Polar and Cylindrical Coordinates

Polar and cylindrical coordinates are useful ways to describe position when rotation or cylindrical shape is involved. Cartesian coordinates work well for displacement, velocity, and forces, but they can become awkward when an object turns or curves around an axis. In the polar coordinate system, a point in a plane is described by two values. The first is the radial coordinate, or the distance from the origin. The second is the angle made by the radial vector, usually measured from the...

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spherical coordinates

Spherical Coordinates for 3D Positioning

Spherical coordinates describe position in three-dimensional space using radius, polar angle, and azimuthal angle. They are especially useful for systems with spherical symmetry, where the shape matches a sphere. Compared with Cartesian coordinates, spherical coordinates can make calculations simpler. To describe a sphere, Cartesian coordinates need three values. Spherical coordinates can use just one parameter, the radius. That is why they are common in science and engineering problems such...

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vector algebra: graphical method

Adding and Subtracting Vectors Graphically

Vectors can be added, subtracted, and multiplied by scalars. The vector sum of two or more vectors is called the resultant vector, or resultant. Geometric rules help find the resultant and its direction in a plane. For two vectors, the parallelogram rule shows how to build the sum. First, move one vector in parallel so both tails start at the same point. Then draw lines through the heads that are parallel to the opposite vector. The diagonal from the shared origin gives the resultant vector. <br...

vector algebra: method of components

Vector Components for Adding Vectors

Vector components make it easier to add, subtract, and multiply vectors. The parallelogram rule and graphical methods can become hard to use when the math gets complex. One option is to draw vectors to scale, as in navigation, and estimate their lengths and directions from the graph. Another option is the method of components. In many situations, the size and direction of each vector are already known, and the goal is to find the resultant, or combined vector, of several vectors. A real-world...

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scalar product (dot product)

Dot Product and Work in Physics

The dot product links two vectors to a single number. That number is called a scalar. In physics, this idea helps describe work and energy. A common example is work done by a force on an object. Force is a vector, and displacement is also a vector. The work is defined as the scalar product of the force vector and the displacement vector. To find the dot product, multiply the magnitudes of the two vectors by the cosine of the angle between them. The angle can be measured from either vector.

vector product (cross product)

Right-Hand Rule for Vector Multiplication

Vector multiplication gives a vector product, also called the cross product. Its magnitude equals the product of the two vectors times the sine of the angle between them. The result points perpendicular to both vectors. The direction of this perpendicular vector is found with the right-hand thumb rule. Because a plane has two possible perpendicular directions, the rule shows which one to choose. To use it, rotate the first vector about the perpendicular line until it lines up with the second...

scalar and vector triple products

Triple Products in Vectors and Volume

Scalar and vector triple products extend the rules for multiplying two vectors to three vectors. A scalar product gives a scalar, while a vector product gives a vector. These same ideas help define two useful triple-product forms. The scalar triple product is the dot product of one vector with the cross product of two others. It produces a scalar quantity. When the vectors are rotated in a cyclic order, the result stays the same, so this form is associative. It also shows the projection of one...

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gradient and del operator

Gradient Direction on Scalar Fields

The gradient helps describe how a scalar field changes in space. It gives both the size and the direction of the greatest rate of change. In physics and mathematics, this makes it a key tool for studying functions and fields. A simple mountain example shows why direction matters. A slope does not mean much unless it is measured in a specific direction. That is why the directional derivative is used. It is a vector that gives the slope and the direction of change. The gradient of a scalar...

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divergence and curl

Vector Fields: Sources, Sinks, and Rotation

Vector fields can spread outward, pull inward, or rotate around a point. Divergence describes the net outward flow of flux from a small volume through a closed surface as the volume gets very small. In simple terms, it shows how much a vector field spreads out from a point. When the flux moves outward, the divergence is positive. A point with positive divergence is often called a source. When the field contracts toward a point, the divergence is negative. That point is often called a sink.

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second derivatives and laplace operator

Using Del Operators to Build Laplacians

The del operator can form important second-order expressions from gradients, divergence, and curl. These combinations matter in mathematics and physics. They help describe how scalar and vector functions change at a deeper level. For a scalar function, the curl of its gradient is one such expression. For a vector function, the divergence of a curl is another. The curl of a gradient and the divergence of a curl are always zero. The divergence of the gradient of a scalar function leads to the...

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Work, circulation, flux, and volume integrals describe how scalar and vector fields add up over a path, surface, or region. For high school science, these ideas help connect math to motion, flow, and stored mass or energy. A line integral for a vector field adds the dot product of a vector function and a tiny displacement vector along a chosen path. If the path is closed, the result is a closed-line integral. The value around a closed path is called the circulation of the vector field. A...

divergence and stokes' theorems

Divergence and Stokes Theorems in Vector Fields

Divergence and Stokes theorems help connect vector fields to the shapes they pass through. A vector field gives a value with direction at each point, such as electric flow or wind flow. These theorems extend Green's theorem to higher dimensions and also generalize the fundamental theorem of calculus. The divergence theorem focuses on the dot product of a vector field. It lets a difficult surface integral become a volume integral, and it can also work in reverse. With this theorem, the rate of...

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key terms and concepts

Explore line, surface, and volume integrals through work, circulation, flux, and density. Clear high school science coverage of scalar and vector fields.

Work, Circulation, and Flux Integrals | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

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Average and Instantaneous Velocity Vectors

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Analyze the motion of an object in terms of its position, velocity, and acceleration (with respect to a frame of reference) as functions of time.

SC.912.P.12.2

Newton's First Law: Introduction

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First Law: Particles in Two-dimensional Equilibrium

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SC.912.P.12.3

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SC.912.P.12.4

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SC.912.P.12.5

A. Motion can be measured and described qualitatively and quantitatively. Net forces create a change in motion. When objects travel at speeds comparable to the speed of light, Einstein's special theory of relativity applies.

B. Momentum is conserved under well-defined conditions. A change in momentum occurs when a net force is applied to an object over a time interval.

C. The Law of Universal Gravitation states that gravitational forces act on all objects irrespective of their size and position.

D. Gases consist of great numbers of molecules moving in all directions. The behavior of gases can be modeled by the kinetic molecular theory.

E. Chemical reaction rates change with conditions under which they occur. Chemical equilibrium is a dynamic state in which forward and reverse processes occur at the same rates.

Work, Circulation, and Flux Integrals

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