Curl of a function and vector field

In summary, the conversation is about a conceptual question on a practice test involving differentiable functions and vector fields. The question asks for the computation of the vector field Curl(fF). The solution involves using the product rule and simplifying to fcurlF + (∇f)XF. The conversation ends with the person thanking for the helpful suggestion.
  • #1
arduinohero
2
1
Hello, I'm having some difficulty with a conceptual question on a practice test I was using to study. I have the answer but not the solution unfortunately.

1. Homework Statement

"For every differentiable function f = f(x,y,z) and differentiable 3-dimensional vector field F=F(x,y,z), the vector field Curl(fF) equals: "

Homework Equations


curlF = ∇XF

The Attempt at a Solution


The solution is apparently: fCurl(F)+∇f x F
I am a little lost the process for this question. I attempted to "solve" the problem using a general case. Essentially I let F = <P, Q, R> and multiplied in "f," so fF = <fP, fQ, fR>.
I then took the curl using the formula. I was left with the following:

curl(fF) = <d/dy(fR)-d/dz(fQ), d/dz(fP)-d/dx(fR), d/dx(fQ)-d/dy(fP)>

I am unsure of where to go from here. I originally was going to factor out "f," but then realized that that is not necessarily possible due to it being within the derivative operator. Any suggestions for a next step would be very helpful!
 
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  • #2
Use the product rule.
 
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Likes arduinohero
  • #3
vela said:
Use the product rule.
I don't know why I didn't think of that... Thank you, that was very helpful!

For anyone interested, I used product rule to differentiate everything and simplified to the following:

<f(Ry-Qz)+fyR-fzQ, f(Pz-Rx)+fzP-fxR, f(Qx-Py)+fxQ-fyP>

I then split the vector into a sum of vectors:

<f(Ry-Qz), f(Pz-Rx), f(Qx-Py)> + <fyR-fzQ, +fzP-fxR, fxQ-fyP>

The first vector being added is equal to fcurlF, and the second is quite clearly <fx, fy, fz> crossed with <P, Q, R>. Since <fx, fy, fz> is in fact ∇f, I rewrote the summation as fcurlF + (∇f)XF, which is of course the solution. Thanks again for the help!
 

1. What is the curl of a function?

The curl of a function is a mathematical operation that describes the rotation or circulation of a vector field. It is a vector quantity that represents the amount and direction of rotation at any given point in the vector field.

2. How is the curl of a function calculated?

The curl of a function is calculated using a specific formula that involves taking the partial derivatives of the function with respect to each coordinate direction. This formula is also known as the "cross product of gradients" or the "vector product of differentials."

3. What does a positive (or negative) curl value indicate?

A positive curl value indicates counterclockwise rotation, while a negative curl value indicates clockwise rotation. In other words, a positive curl value means that the vector field is rotating in a counterclockwise direction at that point, and a negative curl value means it is rotating in a clockwise direction.

4. Can the curl of a function be zero?

Yes, the curl of a function can be zero. This indicates that the vector field is irrotational, meaning that there is no rotation at any point in the field. In other words, the vector field is "curl-free."

5. What are some real-world applications of the curl of a function?

The concept of curl is used in many areas of science and engineering, including fluid dynamics, electromagnetism, and even computer graphics. In fluid dynamics, the curl of a velocity field is used to determine the presence and strength of vortices. In electromagnetism, the curl of the electric and magnetic fields determines the strength and direction of electromagnetic forces. In computer graphics, the curl of a vector field is used to create realistic textures and animations.

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