Geometrical definition of Curl -- proof

In summary, the geometrical definition of curl relates to the rotation or circulation of a vector field around a point. The proof involves visualizing the vector field as a fluid flow and examining the tendency of the fluid to rotate around an infinitesimal loop. By calculating the circulation of the vector field around this loop and taking the limit as the loop shrinks to a point, one can derive the expression for curl in terms of the field’s components. This highlights the relationship between local rotation and the mathematical representation of curl, providing a foundational understanding of its significance in vector calculus.
  • #1
vgarg
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Homework Statement
Need help with understanding proof of geometrical form of Curl.
Relevant Equations
Please see the attachment.
Can someone please explain me the rationale for the terms circled in red on the attached copy of page 400 of "Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering, 3rd edition"?
Thank you.

Mentor Note: approved - it is only a single book page, so no copyright issue.
 

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  • Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering 2006 - pg 400.pdf
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  • #2
vgarg said:
Homework Statement: Need help with understanding proof of geometrical form of Curl.
Relevant Equations: Please see the attachment.

Can someone please explain me the rationale for the terms circled in red on the attached copy of page 400 of "Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering, 3rd edition"?
Thank you.

Mentor Note: approved - it is only a single book page, so no copyright issue.
In curvilinear coordinates, the coordinate lines are not necessarily straight lines, and the scale factors (metric coefficients) ##h_2## and ##h_3## can vary with the coordinates ##u_2## and ##u_3##. Consequently, the components of the vector field ##\mathbf{a}## can change with respect to the coordinates, leading to nonzero derivative terms.

##\frac{\partial}{\partial u_2} (a_3 h_3)## and ##\frac{\partial}{\partial u_3} (a_2 h_2)## account for the the change in the components of the vector field ##\mathbf{a}## as we move along the coordinate directions.

Edit: I'm not an expert on Calculus in curvilinear coordinates, so take my comment with a grain of salt if you can.
 
Last edited:
  • #3
docnet said:
In curvilinear coordinates, the coordinate lines are not necessarily straight lines, and the scale factors (metric coefficients) ##h_2## and ##h_3## can vary with the coordinates ##u_2## and ##u_3##. Consequently, the components of the vector field ##\mathbf{a}## can change with respect to the coordinates, leading to nonzero derivative terms.

##\frac{\partial}{\partial u_2} (a_3 h_3)## and ##\frac{\partial}{\partial u_3} (a_2 h_2)## account for the the change in the components of the vector field ##\mathbf{a}## as we move along the coordinate directions.

Edit: I'm not an expert on Calculus in curvilinear coordinates, so take my comment with a grain of salt if you can.
But why in only 2 directions and not in the other 2 directions?
 
  • #4
I think its because the planar surface ##PQRS## is locally perpendicular to ##\hat{e}_1## in the curvilinear system, so any variations in the ##u_1## direction don't affect the surface defined by the ##u_2## and ##u_3## coordinates.
 
  • #5
Can one of the mentors please move this thread to Mathematics, Calculus forum? May be someone in that form can help me with this. Thank you!
 
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FAQ: Geometrical definition of Curl -- proof

What is the geometrical definition of curl?

The geometrical definition of curl relates to the rotation of a vector field in three-dimensional space. It describes how much and in what direction a vector field "curls" around a point. Mathematically, the curl of a vector field F is a vector that represents the infinitesimal rotation at a point in the field, often visualized as the axis of rotation and the magnitude of the rotation.

How is curl mathematically represented?

The curl of a vector field F, given by F = (P, Q, R), where P, Q, and R are the components of the vector field in the x, y, and z directions respectively, is mathematically represented as: curl F = ∇ x F = (∂R/∂y - ∂Q/∂z, ∂P/∂z - ∂R/∂x, ∂Q/∂x - ∂P/∂y). This expression uses the del operator (∇) and cross product to compute the curl in Cartesian coordinates.

What is the physical interpretation of curl?

The physical interpretation of curl can be understood in terms of fluid flow. If you imagine a small paddle wheel placed in a fluid represented by the vector field, the curl at that point indicates how the wheel would rotate. A positive curl implies counterclockwise rotation, while a negative curl indicates clockwise rotation, with the magnitude representing the strength of the rotation.

How can one visualize curl geometrically?

Curl can be visualized by imagining tiny loops or paddles placed in the vector field. The direction of the curl vector indicates the axis about which the paddles would rotate, and the magnitude indicates how fast they would spin. This visualization helps in understanding how the vector field behaves locally around a point and how it influences the motion of particles within the field.

What are some common applications of curl in physics and engineering?

Curl is commonly used in various fields such as fluid dynamics, electromagnetism, and mechanical engineering. In fluid dynamics, it helps analyze the rotation of fluid elements, while in electromagnetism, curl is used in Maxwell's equations to describe the relationship between electric and magnetic fields. Understanding curl is crucial for analyzing rotational effects in engineering systems, such as the flow of air over wings or the behavior of rotating machinery.

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