The Curl of a Vector Field: Measuring Rotation

Name: The Curl of a Vector Field: Measuring Rotation
Uploaded: 2022-04-21T14:45:02.000Z
Duration: 52 min 49 s

Understanding Curl in Vector Fields

Introduction to Curl

The lecture focuses on the curl of vector fields, a key operation in vector calculus, following previous discussions on gradient and divergence.

The speaker introduces two vector fields: one diverging and another with zero divergence, highlighting their rotational characteristics.

Definition and Calculation of Curl

Curl is defined as a differential operator (del cross product), which indicates how much a vector field is rotating.

The del operator is expressed in three dimensions as partial derivatives with respect to x, y, and z.

A vector-valued function f consists of three components f_1, f_2, and f_3 , each dependent on x, y, and z.

Properties of Curl

The curl of f , denoted as curl(f), results from the cross product between the del operator and the vector function f .

Unlike gradient (scalar to vector) or divergence (vector to scalar), curl takes a vector input and produces another vector output.

Example Calculation

An example function f = (x,y,-sin(z),1) is introduced for practical computation of curl.

The speaker emphasizes understanding both ijk notation and column vector notation for representing vectors interchangeably.

Determinant Computation for Curl

Curl of Vector Fields: Understanding the Basics

Computing the Curl

The determinant for the curl is confirmed to be the cosine of z, with alternating signs applied in calculations for different directions (i, j, k).

The j component results in zero due to both partial derivatives being zero; thus, there is no contribution from this direction.

The final expression for the curl of the vector field is given as textcurl = (cos z, 0, -x) , indicating a straightforward computation method for any vector field.

Interpretation of Curl

When analyzing simpler 2D vector fields represented as f_1, f_2, 0 , it’s noted that all curls will point in the k direction.

The formula for computing curl in 2D is established: textcurl = partial f_2/partial x - partial f_1/partial y , emphasizing its perpendicular nature to the board.

Characteristics of Vector Fields

A vector field with a curl of zero is termed "curl free" or "irrotational," indicating no rotational component present within it.

An intuitive understanding suggests that particles originating at an angle will continue along that path without rotation if a field is irrotational.

Rotational vs. Irrotational Fields

In contrast to irrotational fields, some vector fields exhibit rotation; these are identified by non-zero curl values.

For example, calculating a specific vector field yields a curl value of 2, categorizing it as rotational and demonstrating how divergence can coexist with rotation.

Intuition Behind Curl

It’s explained that vector fields can possess both swirling and expanding characteristics; examples include unstable spiral sources or spiraling inward scenarios.

Positive and negative curls are discussed concerning their directional conventions based on right-hand rules; this highlights how orientation affects interpretation.

Understanding Curl in Fluid Dynamics and Rotational Motion

Fluid Flow and Vector Fields

The concept of fluid flow is introduced, comparing vector fields to those observed in natural scenarios like the Gulf of Mexico or water from a faucet. This analogy helps visualize how vector fields operate intuitively.

Asteroids and Rotational Dynamics

The discussion shifts to an asteroid rotating in space, emphasizing that this scenario can also be interpreted through the lens of curl in vector fields.

Coordinate System and Angular Velocity

A three-dimensional coordinate system (x, y, z) is established for the asteroid's rotation about the z-axis. The angular velocity vector omega is defined as a scalar quantity representing the rate of rotation around this axis.

Clarification is made between the angular velocity vector mathbfw and its scalar representation omega , highlighting their distinct roles in describing rotational motion.

Velocity Induced by Rotation

Each point on the asteroid has a corresponding radial vector r , which defines its position relative to the origin. As the asteroid rotates, these points move with a velocity v , which is tangential to their circular path around axis w .

The relationship between linear velocity v , angular velocity w , and radial position r is expressed mathematically using cross products: v = w times r. This illustrates how each point's motion can be derived from its position relative to the rotation axis.

Calculating Velocity Components

Assuming constant angular velocity along the z-direction, coordinates for point q are specified as combinations of x, y, and z components.

An exercise is suggested where students compute velocities using cross products based on defined vectors. The resulting expression indicates that points aligned with the rotation axis have zero velocity due to their fixed position during rotation.

Characteristics of Rotational Motion

It’s noted that points closer to the axis rotate slower than those farther away; thus establishing a gradient of speed across different distances from the center.

Transitioning into cylindrical coordinates reveals that movement depends solely on radius rather than angle ( theta), indicating uniformity in rotational dynamics regardless of orientation around an axis.

Curl Calculation for Solid Body Rotation

Understanding Curl and Its Properties in Fluid Dynamics

The Concept of Curl in Solid Body Rotation

The curl of a vector field representing particles in a solid body, like an asteroid, is equal to twice the angular rate (omega) about the axis of rotation. This highlights the relationship between curl and rotational motion.

A constant positive curl indicates solid body rotation, meaning that the geometry of the rotating body remains unchanged during its motion. Objects within this flow retain their shape while rotating.

Key Properties of Curl

An important property is that the curl of a gradient always equals zero. This means for any scalar function f , nabla times (nabla f) = 0 . Such fields are termed potential flow solutions and are irrotational.

Similarly, it is established that the divergence of a curl is also zero: nabla cdot (nabla times mathbfF) = 0 . This reinforces that curls do not contribute to divergence.

Understanding Rotational Components

When computing the curl of a vector field, it isolates only those components that exhibit rotational behavior. If there are both swirling and expanding components, only the swirling part will be extracted by taking the curl.